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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span><br/> for <font color="Olive" title="b4">G1</font>, <font color="Olive" title="b5">G2</font> being 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#K13" title="POLYNOM1:func.13">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">G1</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> &amp; <font color="Olive" title="b4">G1</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Olive" title="b1">T</font> &amp; <font color="Olive" title="b5">G2</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> &amp; <font color="Olive" title="b5">G2</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">G2</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span><br/> for <font color="Olive" title="b3">G1</font>, <font color="Olive" title="b4">G2</font> being 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#K13" title="POLYNOM1:func.13">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">G1</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> &amp; <font color="Olive" title="b3">G1</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b4">G2</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> &amp; <font color="Olive" title="b4">G2</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">G2</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span><br/> for <font color="Olive" title="b2">G1</font>, <font color="Olive" title="b3">G2</font> being 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#K13" title="POLYNOM1:func.13">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">G1</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> &amp; <font color="Olive" title="b2">G1</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b3">G2</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> &amp; <font color="Olive" title="b3">G2</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G2</font></span><br/><span class="kw">let </span><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#K13" title="POLYNOM1:func.13">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being 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#K13" title="POLYNOM1:func.13">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">G1</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> &amp; <font color="Olive" title="b1">G1</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Olive" title="b2">G2</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> &amp; <font color="Olive" title="b2">G2</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font>, <font color="Maroon" title="c6">H</font> be  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#K13" title="POLYNOM1:func.13">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c5">G</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c6">H</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> &amp; <font color="Maroon" title="c6">H</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c5">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 class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E3:57"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c6">H</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 class="kw">and </span><br/><a NAME="E4:57"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c6">H</font> <a href="groeb_1.html#R4" title="GROEB_1:pred.4">is_reduced_wrt</a> <font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font></span><br/>

<a NAME="E5:57"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<font color="Maroon" title="c6">H</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">GH</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E6:57"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:57_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><div><span class="lab"><font color="Green" title="E46">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> implies <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_1_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font></span><br/><a NAME="E2:57_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:57_1_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> implies <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_1_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font></span><br/><a NAME="E2:57_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:57_1_2"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_1"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:57_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E3:57_1"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">GH</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font></span>)</span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/>
<div><span class="lab"><font color="Green" title="E45">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font> implies <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:57_2"/><span class="lab"><font color="Green" title="E46">A10</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span>)</span></span><br/><a NAME="E2:57_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A11</font></span>: 
not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:57_2"/><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_2"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:57_2"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font> or <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:57_2_1"/><span class="lab"><font color="Green" title="E49">A13</font></span>: 
not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_2_1_1_1"/>
(  not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> or <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_2_1"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:57_2_1_1_1_1"/>
not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:57_2_1_1_1_1"/>
( not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_2"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:57_2_1_1_1_2"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:57_2_1_1_1_2"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_2"><span class="lab"><font color="Green" title="E47">A11</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:57_2_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:57_2"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">g</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E11:57"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font>
 <span class="kw">and </span><br/><a NAME="E12:57"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
for <font color="Olive" title="b1">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font> holds <br/><font color="Maroon" title="c9">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T31" title="POLYRED:th.31">POLYRED:31</a></span>;<br/>
<a NAME="E13:57"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<font color="Maroon" title="c5">G</font> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E49">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c5">G</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> ) implies ( <font color="Maroon" title="c10">u</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:57_3"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">u</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_3_1_1"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_3"><span class="lab"><font color="Green" title="E50">A18</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_1_1"><span class="kw">case </span></a><a NAME="E1:57_3_1_1_1"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">u</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">u9</font> = <font color="Maroon" title="c10">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">u9</font> = <font color="Maroon" title="c11">u9</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E4:57_3_1_1_1"/>
<font color="Maroon" title="c12">u9</font> <a href="groeb_1.html#R3" title="GROEB_1:pred.3">is_monic_wrt</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E1:57_3_1_1_1"><span class="lab"><font color="Green" title="E51">A19</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>;<br/><a NAME="E5:57_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A20</font></span>: 
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">u9</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#D6" target="_self" title="GROEB_1:def.6">Def6</a></span>;<br/><a NAME="E6:57_3_1_1_1"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 ;<br/><span class="kw">hence </span><a NAME="E7:57_3_1_1_1"/>
( <font color="Maroon" title="c10">u</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_3_1_1_1"><span class="lab"><font color="Green" title="E52">A20</font></span></a>, <a class="ref" href="termord.html#T17" title="TERMORD:th.17">TERMORD:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1_1_2"><span class="kw">case </span></a><a NAME="E1:57_3_1_1_2"/><span class="lab"><font color="Green" title="E51">A21</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">u</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">u9</font> = <font color="Maroon" title="c10">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">u9</font> = <font color="Maroon" title="c11">u9</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E4:57_3_1_1_2"/>
<font color="Maroon" title="c12">u9</font> <a href="groeb_1.html#R3" title="GROEB_1:pred.3">is_monic_wrt</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E1:57_3_1_1_2"><span class="lab"><font color="Green" title="E51">A21</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>;<br/><a NAME="E5:57_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A22</font></span>: 
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c12">u9</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#D6" target="_self" title="GROEB_1:def.6">Def6</a></span>;<br/><a NAME="E6:57_3_1_1_2"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 ;<br/><span class="kw">hence </span><a NAME="E7:57_3_1_1_2"/>
( <font color="Maroon" title="c10">u</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57_3_1_1_2"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="termord.html#T17" title="TERMORD:th.17">TERMORD:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:57_3"/>
( <font color="Maroon" title="c10">u</font> is   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:57"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c2">T</font> is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</a></span>;<br/>
<a NAME="E16:57"/><span class="kw">then </span>
for <font color="Olive" title="b1">f</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">f</font>, <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T15" target="_self" title="GROEB_1:th.15">Th15</a></span>;<br/>
<a NAME="E17:57"/><span class="kw">then </span>
for <font color="Olive" title="b1">f</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T16" target="_self" title="GROEB_1:th.16">Th16</a></span>;<br/>
<a NAME="E18:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A23</font></span>: 
for <font color="Olive" title="b1">f</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T17" target="_self" title="GROEB_1:th.17">Th17</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E51">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">u</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c5">G</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> ) implies  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_4"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span><br/><a NAME="E2:57_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">u</font> <a href="groeb_1.html#R3" title="GROEB_1:pred.3">is_monic_wrt</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:57_4"/>
 <a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">u</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#D6" target="_self" title="GROEB_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E52">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_5"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> )</span><br/><a NAME="E2:57_5"/><span class="kw">then </span>
not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:57_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A27</font></span>: 
(  not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> or  not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E4:57_5"/><span class="lab"><font color="Green" title="E55">A28</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_5"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_5_1_1"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_5"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_5_1_1_1"><span class="kw">case </span></a><a NAME="E1:57_5_1_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:57_5_1_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_5"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_5_1_1_2"><span class="kw">case </span></a><a NAME="E1:57_5_1_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:57_5_1_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_5"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:57_5"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span>)</span> implies <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font> )</span><br/><span class="kw">assume </span><a NAME="E1:57_6"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c8">GH</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_6_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_6_1_1"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font> or <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_6"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_6_1_1_1"><span class="kw">case </span></a><a NAME="E1:57_6_1_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font> )</span><br/><div class="add"><a NAME="E2:57_6_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:57_6_1_1_1"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_6_1_1_2"><span class="kw">case </span></a><a NAME="E1:57_6_1_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font> )</span><br/><div class="add"><a NAME="E2:57_6_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:57_6_1_1_2"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c6">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:57_6"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A30</font></span>: 
<font color="Maroon" title="c8">GH</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/>
<a NAME="E23:57"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">H</font>,<font color="Maroon" title="c2">T</font> is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="groeb_1.html#D4" target="_self" title="GROEB_1:def.4">Def4</a></span>;<br/>
<a NAME="E24:57"/><span class="kw">then </span>
for <font color="Olive" title="b1">f</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/> <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">H</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Olive" title="b1">f</font>, <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T15" target="_self" title="GROEB_1:th.15">Th15</a></span>;<br/>
<a NAME="E25:57"/><span class="kw">then </span>
for <font color="Olive" title="b1">f</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c6">H</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T16" target="_self" title="GROEB_1:th.16">Th16</a></span>;<br/>
<a NAME="E26:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A31</font></span>: 
for <font color="Olive" title="b1">f</font> being  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="polyred.html#R10" title="POLYRED:pred.10">is_top_reducible_wrt</a> <font color="Maroon" title="c6">H</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T17" target="_self" title="GROEB_1:th.17">Th17</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_7"/>
( <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font> or <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:57"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E11:57"><span class="lab"><font color="Green" title="E46">A14</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_1"><span class="kw">suppose </span></a><a NAME="E1:57_7_1"/><span class="lab"><font color="Green" title="E55">A32</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A33</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A34</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E5:57_7_1"/><span class="lab"><font color="Green" title="E58">A35</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D15" title="IDEAL_1:def.15">IDEAL_1:def 15</a></span>;<br/><a NAME="E6:57_7_1"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E2:57_7_1"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E3:57_7_1"><span class="lab"><font color="Green" title="E57">A34</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b9</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E8:57_7_1"/><span class="lab"><font color="Green" title="E59">A36</font></span>: 
<font color="Maroon" title="c11">b9</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="c6">H</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E9:57_7_1"/><span class="lab"><font color="Green" title="E60">A37</font></span>: 
<font color="Maroon" title="c11">b9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:57"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="ref" href="groeb_1.html#T18" target="_self" title="GROEB_1:th.18">Th18</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">h</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E11:57_7_1"/><span class="lab"><font color="Green" title="E61">A38</font></span>: 
<font color="Maroon" title="c11">b9</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="c12">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E12:57_7_1"/><span class="lab"><font color="Green" title="E62">A39</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">and </span><br/><a NAME="E13:57_7_1"/><span class="lab"><font color="Green" title="E63">A40</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_7_1"><span class="lab"><font color="Green" title="E59">A36</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c12">h</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_7_1"><span class="lab"><font color="Green" title="E63">A40</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c10">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <font color="Maroon" title="c13">h</font>;<br/><a NAME="E15:57_7_1"/><span class="lab"><font color="Green" title="E64">A41</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_1"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="txt" href="#E12:57_7_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E65">A42</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_7_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_7_1_1"/><span class="lab"><font color="Green" title="E66">A43</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><font color="Maroon" title="c13">h</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c13">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c13">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_1_1"><span class="lab"><font color="Green" title="E66">A43</font></span></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c13">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></span>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:57_7_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_7_1"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:57_7_1"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_7_1"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/><a NAME="E18:57_7_1"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><a NAME="E19:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A44</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_7_1"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="txt" href="#E11:57_7_1"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="ref" href="polyred.html#T36" title="POLYRED:th.36">POLYRED:36</a></span>;<br/><a NAME="E20:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A45</font></span>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <font color="Maroon" title="c10">g</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_7_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_7_1_2"/>
not <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:57_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A46</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_7_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:57_7_1_2"/>
not <font color="Maroon" title="c13">h</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"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_7_1"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:57_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_7_1_2"><span class="lab"><font color="Green" title="E68">A46</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">r</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E6:57_7_1_2"/><span class="lab"><font color="Green" title="E69">A47</font></span>: 
( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:57_7_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a></span>;<br/><a NAME="E7:57_7_1_2"/><span class="lab"><font color="Green" title="E70">A48</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_7_1_2"><span class="lab"><font color="Green" title="E69">A47</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E8:57_7_1_2"/>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#NR10" title="POLYRED:NR.10">is_irreducible_wrt</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E2:57_7_1"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:57_7_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_7_1_2"><span class="lab"><font color="Green" title="E70">A48</font></span></a>, <a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E22:57_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:57"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E23:57_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57"><span class="lab"><font color="Green" title="E47">A15</font></span></a></span>;<br/><a NAME="E24:57_7_1"/><span class="kw">then </span>
not <font color="Maroon" title="c13">h</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T29" title="POLYRED:th.29">POLYRED:29</a></span>;<br/><a NAME="E25:57_7_1"/><span class="kw">then </span>
not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T32" title="POLYRED:th.32">POLYRED:32</a></span>;<br/><a NAME="E26:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A49</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></span>;<br/><a NAME="E27:57_7_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57_7_1"><span class="lab"><font color="Green" title="E66">A44</font></span></a>, <a class="ref" href="groeb_1.html#T9" target="_self" title="GROEB_1:th.9">Th9</a></span>;<br/><a NAME="E28:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A50</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</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="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:57_7_1"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c10">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <font color="Maroon" title="c13">h</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57_7_1"><span class="lab"><font color="Green" title="E65">A42</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E30:57_7_1"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57_7_1"><span class="lab"><font color="Green" title="E65">A42</font></span></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/><a NAME="E31:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A51</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_7_1"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D7" title="TERMORD:def.7">TERMORD:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D7" title="TERMORD:def.7">TERMORD:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E2:57_7_1"><span class="lab"><font color="Green" title="E56">A33</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E12:57_7_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
;<br/><a NAME="E33:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A52</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:57_7_1"><span class="lab"><font color="Green" title="E70">A51</font></span></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/><a NAME="E34:57_7_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T7" target="_self" title="GROEB_1:th.7">Th7</a></span>;<br/><a NAME="E35:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A53</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_7_1"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="termord.html#T12" title="TERMORD:th.12">TERMORD:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">g9</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c17">h9</font> = <font color="Maroon" title="c13">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">g9</font> = <font color="Maroon" title="c16">g9</font>, <font color="Maroon" title="c19">h9</font> = <font color="Maroon" title="c17">h9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><a NAME="E38:57_7_1"/>
<font color="Maroon" title="c6">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">H</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D15" title="IDEAL_1:def.15">IDEAL_1:def 15</a></span>;<br/><a NAME="E39:57_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">g9</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E2:57_7_1"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E5:57_7_1"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="txt" href="#E12:57_7_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="ideal_1.html#T15" title="IDEAL_1:th.15">IDEAL_1:15</a></span>;<br/><a NAME="E40:57_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="groeb_1.html#E4"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/><a NAME="E41:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A54</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57_7_1"><span class="lab"><font color="Green" title="E65">A42</font></span></a></span>;<br/><a NAME="E42:57_7_1"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T79" title="POLYNOM1:th.79">POLYNOM1:79</a></span>;<br/><a NAME="E43:57_7_1"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>;<br/><a NAME="E44:57_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A55</font></span>: 
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c13">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T5" target="_self" title="GROEB_1:th.5">Th5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_7_1_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_7_1_4_1"/>
(  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font> or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c13">h</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:57_7_1"><span class="lab"><font color="Green" title="E70">A51</font></span></a>, <a class="txt" href="#E44:57_7_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_1_4_1_1"><span class="kw">case </span></a><a NAME="E1:57_7_1_4_1_1"/><span class="lab"><font color="Green" title="E75">A56</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c20">b9</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:57_7_1_4_1_1"/><span class="lab"><font color="Green" title="E76">A57</font></span>: 
<font color="Maroon" title="c20">b9</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="c5">G</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E4:57_7_1_4_1_1"/><span class="lab"><font color="Green" title="E77">A58</font></span>: 
<font color="Maroon" title="c20">b9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E18:57"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E41:57_7_1"><span class="lab"><font color="Green" title="E73">A54</font></span></a>, <a class="ref" href="groeb_1.html#T18" target="_self" title="GROEB_1:th.18">Th18</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E6:57_7_1_4_1_1"/><span class="lab"><font color="Green" title="E78">A59</font></span>: 
<font color="Maroon" title="c20">b9</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="c21">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E7:57_7_1_4_1_1"/><span class="lab"><font color="Green" title="E79">A60</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E8:57_7_1_4_1_1"/><span class="lab"><font color="Green" title="E80">A61</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E76">A57</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c21">q</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E80">A61</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E10:57_7_1_4_1_1"/>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E75">A56</font></span></a>, <a class="txt" href="#E4:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E77">A58</font></span></a>, <a class="txt" href="#E6:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E78">A59</font></span></a>, <a class="ref" href="polyred.html#T36" title="POLYRED:th.36">POLYRED:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E12:57_7_1_4_1_1"/><span class="lab"><font color="Green" title="E81">A62</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><a NAME="E13:57_7_1_4_1_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E77">A58</font></span></a>, <a class="txt" href="#E6:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E78">A59</font></span></a>, <a class="ref" href="termord.html#T10" title="TERMORD:th.10">TERMORD:10</a></span>;<br/><a NAME="E14:57_7_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:57_7_1"><span class="lab"><font color="Green" title="E72">A53</font></span></a>, <a class="txt" href="#E33:57_7_1"><span class="lab"><font color="Green" title="E71">A52</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>;<br/><a NAME="E15:57_7_1_4_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c22">q</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"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E16:57_7_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E79">A60</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E17:57_7_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_7_1_4_1_1"><span class="lab"><font color="Green" title="E81">A62</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E18:57_7_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E19:57_7_1_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E2:57_7_1"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_1_4_1_2"><span class="kw">case </span></a><a NAME="E1:57_7_1_4_1_2"/><span class="lab"><font color="Green" title="E75">A63</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c13">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c20">b9</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:57_7_1_4_1_2"/><span class="lab"><font color="Green" title="E76">A64</font></span>: 
<font color="Maroon" title="c20">b9</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="c6">H</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E4:57_7_1_4_1_2"/><span class="lab"><font color="Green" title="E77">A65</font></span>: 
<font color="Maroon" title="c20">b9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E26:57"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E41:57_7_1"><span class="lab"><font color="Green" title="E73">A54</font></span></a>, <a class="ref" href="groeb_1.html#T18" target="_self" title="GROEB_1:th.18">Th18</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E6:57_7_1_4_1_2"/><span class="lab"><font color="Green" title="E78">A66</font></span>: 
<font color="Maroon" title="c20">b9</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="c21">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E7:57_7_1_4_1_2"/><span class="lab"><font color="Green" title="E79">A67</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">and </span><br/><a NAME="E8:57_7_1_4_1_2"/><span class="lab"><font color="Green" title="E80">A68</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E76">A64</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c21">q</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E80">A68</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E10:57_7_1_4_1_2"/>
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E75">A63</font></span></a>, <a class="txt" href="#E4:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="txt" href="#E6:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="ref" href="polyred.html#T36" title="POLYRED:th.36">POLYRED:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E12:57_7_1_4_1_2"/><span class="lab"><font color="Green" title="E81">A69</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><a NAME="E13:57_7_1_4_1_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="txt" href="#E6:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="ref" href="termord.html#T10" title="TERMORD:th.10">TERMORD:10</a></span>;<br/><a NAME="E14:57_7_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_7_1"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E35:57_7_1"><span class="lab"><font color="Green" title="E72">A53</font></span></a>, <a class="txt" href="#E33:57_7_1"><span class="lab"><font color="Green" title="E71">A52</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>;<br/><a NAME="E15:57_7_1_4_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c22">q</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"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E16:57_7_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E79">A67</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E17:57_7_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_7_1_4_1_2"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E18:57_7_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E19:57_7_1_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E12:57_7_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E46:57_7_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_2"><span class="kw">suppose </span></a><a NAME="E1:57_7_2"/><span class="lab"><font color="Green" title="E55">A70</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A71</font></span>: 
not <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:57_7_2"/><span class="lab"><font color="Green" title="E57">A72</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_2"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A73</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c9">g</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E6:57_7_2"/><span class="lab"><font color="Green" title="E59">A74</font></span>: 
<font color="Maroon" title="c6">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">H</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D15" title="IDEAL_1:def.15">IDEAL_1:def 15</a></span>;<br/><a NAME="E7:57_7_2"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span>)</span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E3:57_7_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E4:57_7_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">b9</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E9:57_7_2"/><span class="lab"><font color="Green" title="E60">A75</font></span>: 
<font color="Maroon" title="c11">b9</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="c5">G</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E10:57_7_2"/><span class="lab"><font color="Green" title="E61">A76</font></span>: 
<font color="Maroon" title="c11">b9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:57"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="ref" href="groeb_1.html#T18" target="_self" title="GROEB_1:th.18">Th18</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">h</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E12:57_7_2"/><span class="lab"><font color="Green" title="E62">A77</font></span>: 
<font color="Maroon" title="c11">b9</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="c12">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E13:57_7_2"/><span class="lab"><font color="Green" title="E63">A78</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E14:57_7_2"/><span class="lab"><font color="Green" title="E64">A79</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:57_7_2"><span class="lab"><font color="Green" title="E60">A75</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c12">h</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:57_7_2"><span class="lab"><font color="Green" title="E64">A79</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c10">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <font color="Maroon" title="c13">h</font>;<br/><div><span class="lab"><font color="Green" title="E65">A80</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_7_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_7_2_1"/><span class="lab"><font color="Green" title="E66">A81</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><font color="Maroon" title="c13">h</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c13">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T2" title="POLYRED:th.2">POLYRED:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c13">h</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_2_1"><span class="lab"><font color="Green" title="E66">A81</font></span></a>, <a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <font color="Maroon" title="c13">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T80" title="POLYNOM1:th.80">POLYNOM1:80</a></span>
<br/>.= 
<font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T3" title="POLYRED:th.3">POLYRED:3</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:57_7_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_7_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E13:57_7_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="polynom1.html#T82" title="POLYNOM1:th.82">POLYNOM1:82</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:57_7_2"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_7_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/><a NAME="E18:57_7_2"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><a NAME="E19:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A82</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:57_7_2"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E12:57_7_2"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="polyred.html#T36" title="POLYRED:th.36">POLYRED:36</a></span>;<br/><a NAME="E20:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A83</font></span>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> st <font color="Maroon" title="c10">g</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Olive" title="b1">r</font>,<font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_7_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:57_7_2_2"/>
not <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:57_7_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A84</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57_7_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:57_7_2_2"/>
not <font color="Maroon" title="c13">h</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"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_7_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E13:57_7_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:57_7_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57_7_2_2"><span class="lab"><font color="Green" title="E68">A84</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">r</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E6:57_7_2_2"/><span class="lab"><font color="Green" title="E69">A85</font></span>: 
( <font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:57_7_2"><span class="lab"><font color="Green" title="E67">A83</font></span></a></span>;<br/><a NAME="E7:57_7_2_2"/><span class="lab"><font color="Green" title="E70">A86</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c15">r</font>,<font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:57_7_2_2"><span class="lab"><font color="Green" title="E69">A85</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E8:57_7_2_2"/>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#NR10" title="POLYRED:NR.10">is_irreducible_wrt</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E3:57_7_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:57_7_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_7_2_2"><span class="lab"><font color="Green" title="E70">A86</font></span></a>, <a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E22:57_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">GH</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:57"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E23:57_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R1" title="POLYRED:pred.1">&lt;=</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57"><span class="lab"><font color="Green" title="E47">A15</font></span></a></span>;<br/><a NAME="E24:57_7_2"/><span class="kw">then </span>
not <font color="Maroon" title="c13">h</font> <a href="polyred.html#R2" title="POLYRED:pred.2">&lt;</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T29" title="POLYRED:th.29">POLYRED:29</a></span>;<br/><a NAME="E25:57_7_2"/><span class="kw">then </span>
not  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T32" title="POLYRED:th.32">POLYRED:32</a></span>;<br/><a NAME="E26:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A87</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></span>;<br/><a NAME="E27:57_7_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57_7_2"><span class="lab"><font color="Green" title="E66">A82</font></span></a>, <a class="ref" href="groeb_1.html#T9" target="_self" title="GROEB_1:th.9">Th9</a></span>;<br/><a NAME="E28:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A88</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</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="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:57_7_2"><span class="lab"><font color="Green" title="E68">A87</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c10">g</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-</a> <font color="Maroon" title="c13">h</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57_7_2"><span class="lab"><font color="Green" title="E65">A80</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E30:57_7_2"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57_7_2"><span class="lab"><font color="Green" title="E65">A80</font></span></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/><a NAME="E31:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A89</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><font color="Maroon" title="c15">f</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_7_2"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="ref" href="polynom1.html#D22" title="POLYNOM1:def.22">POLYNOM1:def 22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D7" title="TERMORD:def.7">TERMORD:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#D7" title="TERMORD:def.7">TERMORD:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E3:57_7_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:57"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="txt" href="#E13:57_7_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
;<br/><a NAME="E33:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A90</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:57_7_2"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/><a NAME="E34:57_7_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K2" title="TERMORD:func.2">max</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T7" target="_self" title="GROEB_1:th.7">Th7</a></span>;<br/><a NAME="E35:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A91</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_7_2"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="ref" href="termord.html#T12" title="TERMORD:th.12">TERMORD:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">g9</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c17">h9</font> = <font color="Maroon" title="c13">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">g9</font> = <font color="Maroon" title="c16">g9</font>, <font color="Maroon" title="c19">h9</font> = <font color="Maroon" title="c17">h9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span> ;<br/><a NAME="E38:57_7_2"/>
<font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D15" title="IDEAL_1:def.15">IDEAL_1:def 15</a></span>;<br/><a NAME="E39:57_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">g9</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c19">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E3:57_7_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E6:57_7_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E13:57_7_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="ideal_1.html#T15" title="IDEAL_1:th.15">IDEAL_1:15</a></span>;<br/><a NAME="E40:57_7_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="groeb_1.html#E4"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/><a NAME="E41:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A92</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:57_7_2"><span class="lab"><font color="Green" title="E65">A80</font></span></a></span>;<br/><a NAME="E42:57_7_2"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T79" title="POLYNOM1:th.79">POLYNOM1:79</a></span>;<br/><a NAME="E43:57_7_2"/><span class="kw">then </span>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D23" title="POLYNOM1:def.23">POLYNOM1:def 23</a></span>;<br/><a NAME="E44:57_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A93</font></span>: 
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c13">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#T5" target="_self" title="GROEB_1:th.5">Th5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_7_2_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:57_7_2_4_1"/>
(  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font> or  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c13">h</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:57_7_2"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="txt" href="#E44:57_7_2"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_2_4_1_1"><span class="kw">case </span></a><a NAME="E1:57_7_2_4_1_1"/><span class="lab"><font color="Green" title="E75">A94</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c10">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c20">b9</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:57_7_2_4_1_1"/><span class="lab"><font color="Green" title="E76">A95</font></span>: 
<font color="Maroon" title="c20">b9</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="c6">H</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E4:57_7_2_4_1_1"/><span class="lab"><font color="Green" title="E77">A96</font></span>: 
<font color="Maroon" title="c20">b9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:57"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E26:57"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E41:57_7_2"><span class="lab"><font color="Green" title="E73">A92</font></span></a>, <a class="ref" href="groeb_1.html#T18" target="_self" title="GROEB_1:th.18">Th18</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E6:57_7_2_4_1_1"/><span class="lab"><font color="Green" title="E78">A97</font></span>: 
<font color="Maroon" title="c20">b9</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="c21">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E7:57_7_2_4_1_1"/><span class="lab"><font color="Green" title="E79">A98</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">and </span><br/><a NAME="E8:57_7_2_4_1_1"/><span class="lab"><font color="Green" title="E80">A99</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c21">q</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E80">A99</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E10:57_7_2_4_1_1"/>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E75">A94</font></span></a>, <a class="txt" href="#E4:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E77">A96</font></span></a>, <a class="txt" href="#E6:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E78">A97</font></span></a>, <a class="ref" href="polyred.html#T36" title="POLYRED:th.36">POLYRED:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E12:57_7_2_4_1_1"/><span class="lab"><font color="Green" title="E81">A100</font></span>: 
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><a NAME="E13:57_7_2_4_1_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E77">A96</font></span></a>, <a class="txt" href="#E6:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E78">A97</font></span></a>, <a class="ref" href="termord.html#T10" title="TERMORD:th.10">TERMORD:10</a></span>;<br/><a NAME="E14:57_7_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:57_7_2"><span class="lab"><font color="Green" title="E72">A91</font></span></a>, <a class="txt" href="#E33:57_7_2"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>;<br/><a NAME="E15:57_7_2_4_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c22">q</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"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E16:57_7_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E79">A98</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E17:57_7_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_7_2_4_1_1"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E18:57_7_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">g</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c6">H</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c10">g</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E19:57_7_2_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E3:57_7_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_7_2_4_1_2"><span class="kw">case </span></a><a NAME="E1:57_7_2_4_1_2"/><span class="lab"><font color="Green" title="E75">A101</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c13">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c20">b9</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:57_7_2_4_1_2"/><span class="lab"><font color="Green" title="E76">A102</font></span>: 
<font color="Maroon" title="c20">b9</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="c5">G</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E4:57_7_2_4_1_2"/><span class="lab"><font color="Green" title="E77">A103</font></span>: 
<font color="Maroon" title="c20">b9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:57"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E18:57"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E41:57_7_2"><span class="lab"><font color="Green" title="E73">A92</font></span></a>, <a class="ref" href="groeb_1.html#T18" target="_self" title="GROEB_1:th.18">Th18</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E6:57_7_2_4_1_2"/><span class="lab"><font color="Green" title="E78">A104</font></span>: 
<font color="Maroon" title="c20">b9</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="c21">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E7:57_7_2_4_1_2"/><span class="lab"><font color="Green" title="E79">A105</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E8:57_7_2_4_1_2"/><span class="lab"><font color="Green" title="E80">A106</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K9" title="POLYNOM1:func.9">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E76">A102</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c21">q</font> as   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E80">A106</font></span></a>, <a class="ref" href="polynom7.html#D2" title="POLYNOM7:def.2">POLYNOM7:def 2</a></span>;<br/><a NAME="E10:57_7_2_4_1_2"/>
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E75">A101</font></span></a>, <a class="txt" href="#E4:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E77">A103</font></span></a>, <a class="txt" href="#E6:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E78">A104</font></span></a>, <a class="ref" href="polyred.html#T36" title="POLYRED:th.36">POLYRED:36</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><span class="kw"> such that </span><br/><a NAME="E12:57_7_2_4_1_2"/><span class="lab"><font color="Green" title="E81">A107</font></span>: 
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><a NAME="E13:57_7_2_4_1_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c15">f</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E77">A103</font></span></a>, <a class="txt" href="#E6:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E78">A104</font></span></a>, <a class="ref" href="termord.html#T10" title="TERMORD:th.10">TERMORD:10</a></span>;<br/><a NAME="E14:57_7_2_4_1_2"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c2">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> <font color="Maroon" title="c13">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:57_7_2"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E35:57_7_2"><span class="lab"><font color="Green" title="E72">A91</font></span></a>, <a class="txt" href="#E33:57_7_2"><span class="lab"><font color="Green" title="E71">A90</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>;<br/><a NAME="E15:57_7_2_4_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c22">q</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"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E16:57_7_2_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E79">A105</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E17:57_7_2_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c23">r</font>,<font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:57_7_2_4_1_2"><span class="lab"><font color="Green" title="E81">A107</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E18:57_7_2_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">h</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c5">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="groeb_1.html#K1" title="GROEB_1:func.1">{</a><span class="default"><font color="Maroon" title="c13">h</font></span><a href="groeb_1.html#K1" title="GROEB_1:func.1">}</a></span>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D9" title="POLYRED:def.9">POLYRED:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E19:57_7_2_4_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E13:57_7_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="groeb_1.html#D7" target="_self" title="GROEB_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E46:57_7_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
