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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#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>   holds <br/> not  for <font color="Olive" title="b3">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font>)</span>)</span> holds  <font color="Olive" title="b3">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Olive" title="b1">T</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="firebrick">::  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#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>   holds <br/> not  for <font color="Olive" title="b2">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>)</span>)</span> holds  <font color="Olive" title="b2">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</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#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="firebrick">::  thesis: </font></span></a><span class="hide">  not  for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> holds  <font color="Olive" title="b1">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">1bag</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">1bag</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,1) as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">p</font> = <span class="p1">(<span class="default"><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> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<font color="Maroon" title="c5">1bag</font>,<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>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <span class="p1">(<span class="default"><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> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<font color="Maroon" title="c5">1bag</font>,<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>) as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">L</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ;<br/>
<a NAME="E4:42"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <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">set </span><font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<font color="Maroon" title="c5">1bag</font>,<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>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<font color="Maroon" title="c5">1bag</font>,<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>) as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c3">L</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c10">q</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> ;<br/>
<div><span class="lab"><font color="Green" title="E31">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">u</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">1bag</font> implies <font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:42_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">1bag</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span><br/><a NAME="E3:42_1"/>
<font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/><a NAME="E4:42_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K8" title="POLYNOM1:func.8">1_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T20" title="POLYNOM7:th.20">POLYNOM7:20</a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_1"/>
<font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="polynom1.html#T25" title="POLYNOM1:th.25">POLYNOM1:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E32">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">u9</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font> implies <font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
<font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">u</font> = <font color="Maroon" title="c12">u9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E3:42_2"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">u</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_2_1"/>
 not <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c13">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c13">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">1bag</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E34">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:42_2"/>
<font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:42"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T14" title="ORDINAL1:th.14">ORDINAL1:14</a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E10:42"/><span class="kw">then </span>
<font color="Maroon" title="c5">1bag</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E11:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">1bag</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: <font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T19" title="POLYNOM7:th.19">POLYNOM7:19</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="polynom1.html#T22" title="POLYNOM1:th.22">POLYNOM1:22</a></span>
;<br/>
<div><span class="lab"><font color="Green" title="E35">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">u</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">1bag</font> implies <font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_4"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">1bag</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span><br/><a NAME="E2:42_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/><a NAME="E3:42_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T19" title="POLYNOM7:th.19">POLYNOM7:19</a></span>;<br/><span class="kw">hence </span><a NAME="E4:42_4"/>
<font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="polynom1.html#T22" title="POLYNOM1:th.22">POLYNOM1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E36">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">u9</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c11">q</font> implies <font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:42_5"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c11">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">u</font> = <font color="Maroon" title="c12">u9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E3:42_5"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">u</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_5"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_5_1"/>
 not <font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">1bag</font>
 <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/><span class="kw">hence </span><a NAME="E3:42_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:42"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E3:42_5"><span class="lab"><font color="Green" title="E38">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:42_5"/>
<font color="Maroon" title="c12">u9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:42"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</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="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E17:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">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="polynom1.html#T22" title="POLYNOM1:th.22">POLYNOM1:22</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c11">q</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_6"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c11">q</font></span><br/><a NAME="E2:42_6"/><span class="kw">then </span>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">1bag</font>
 <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/><span class="kw">hence </span><a NAME="E3:42_6"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c11">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A16</font></span>: 
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">q</font> = <font color="Maroon" title="c11">q</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#D4" title="POLYNOM1:def.4">POLYNOM1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">q</font> = <font color="Maroon" title="c12">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="#E17:42"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">q1</font> = <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span>;<br/>
<a NAME="E22:42"/>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c13">q</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="#E17:42"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/>
<a NAME="E23:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A17</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c13">q</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="E24:42"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <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="c13">q</font>,<font color="Maroon" title="c2">T</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T53" title="PRE_POLY:th.53">PRE_POLY:53</a></span>;<br/>
<a NAME="E25:42"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">q</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">q</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="#E17:42"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E23:42"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/>
<a NAME="E26:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></span>;<br/>
<span class="lab"><font color="Green" title="E42">A19</font></span>: <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c13">q</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/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c13">q</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c13">q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c13">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T17" title="POLYRED:th.17">POLYRED:17</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c13">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T27" title="POLYNOM7:th.27">POLYNOM7:27</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</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#K8" title="POLYNOM1:func.8">1_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c13">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T20" title="POLYNOM7:th.20">POLYNOM7:20</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c13">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T30" title="POLYNOM1:th.30">POLYNOM1:30</a></span>
<br/>.= 
 <a href="polynom1.html#K7" title="POLYNOM1:func.7">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="polynom1.html#T24" title="POLYNOM1:th.24">POLYNOM1:24</a></span>
;<br/>
<a NAME="E28:42"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><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> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E29:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A21</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</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="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>;<br/>
<a NAME="E30:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A22</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">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="#E4:42"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="polynom1.html#T22" title="POLYNOM1:th.22">POLYNOM1:22</a></span>;<br/>
<span class="lab"><font color="Green" title="E46">A23</font></span>: <font color="Maroon" title="c8">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> = 
<span class="p1">(<span class="default"><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> <a href="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom1.html#K8" title="POLYNOM1:func.8">1_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T20" title="POLYNOM7:th.20">POLYNOM7:20</a></span>
<br/>.= 
 <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="polynom1.html#T25" title="POLYNOM1:th.25">POLYNOM1:25</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_9"/><span class="lab"><font color="Green" title="E47">A24</font></span>: 
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_9_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_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="E1:42_9_1_1"/>
( <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">1bag</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_9"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_1_1"><span class="kw">case </span></a><a NAME="E1:42_9_1_1_1"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_1_1"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E31:42"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_9_1_1_2"><span class="kw">case </span></a><a NAME="E1:42_9_1_1_2"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">1bag</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_9_1_1_2"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E29:42"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:42_9"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E33:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A25</font></span>: 
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c8">p</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#D4" title="POLYNOM1:def.4">POLYNOM1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">p</font> = <font color="Maroon" title="c15">p</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="#E30:42"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/>
<a NAME="E36:42"/><span class="lab"><font color="Green" title="E48">A26</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <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="c16">p</font>,<font color="Maroon" title="c2">T</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T53" title="PRE_POLY:th.53">PRE_POLY:53</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E49">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_10"><span class="kw">now </span></a><div class="add"><a NAME="E1:42_10"/><span class="lab"><font color="Green" title="E50">A28</font></span>: 
 <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c5">1bag</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="termord.html#T9" title="TERMORD:th.9">TERMORD:9</a></span>;<br/><span class="kw">assume </span><a NAME="E2:42_10"/><span class="lab"><font color="Green" title="E51">A29</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:42_10"/>
<font color="Maroon" title="c5">1bag</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c16">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E4:42_10"/><span class="kw">then </span>
<font color="Maroon" title="c5">1bag</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_10"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="termord.html#D6" title="TERMORD:def.6">TERMORD:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_10"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E1:42_10"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="termord.html#T7" title="TERMORD:th.7">TERMORD:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c17">p1</font> = <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>;<br/>
<a NAME="E38:42"/>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c16">p</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="#E30:42"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="ref" href="polynom7.html#T1" title="POLYNOM7:th.1">POLYNOM7:1</a></span>;<br/>
<a NAME="E39:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A30</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c16">p</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="E40:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A31</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">1bag</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E37:42"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E41:42"/><span class="kw">then </span>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c13">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E42:42"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">p</font>, <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</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="#E30:42"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E36:42"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="ref" href="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/>
<a NAME="E43:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A32</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E53">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_11"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_11"/>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c16">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_11"/><span class="kw">then </span>
 <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E54">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_12"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_12"/>
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">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="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><font color="Maroon" title="c16">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <font color="Maroon" title="c16">p</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="c13">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <font color="Maroon" title="c16">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <font color="Maroon" title="c16">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T21" title="POLYNOM1:th.21">POLYNOM1:21</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">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="E3:42_12"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:42"><span class="lab"><font color="Green" title="E53">A33</font></span></a>, <a class="ref" href="polynom1.html#T23" title="POLYNOM1:th.23">POLYNOM1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c18">P</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c13">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_13"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c13">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> implies <font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:42_13"/>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c13">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span></span><br/><a NAME="E2:42_13"/><span class="kw">then </span>
( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font> or <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_13"/>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">P</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c13">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">P</font> = <font color="Maroon" title="c19">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">R</font> =  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>);<br/>
<span class="kw">take </span>
<font color="Maroon" title="c20">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c20">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font></span><br/>

<a NAME="E49:42"/><span class="lab"><font color="Green" title="E55">A35</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E50:42"/>
<font color="Maroon" title="c13">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E51:42"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>),<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:42"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E27:42"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/>
<a NAME="E52:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A36</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">q</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></span>;<br/>
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p4">(<span class="default"><a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</font></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E40:42"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="ref" href="termord.html#D7" title="TERMORD:def.7">TERMORD:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p3">(<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> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:42"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<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="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D10" title="VECTSP_1:def.10">VECTSP_1:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</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="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="polynom7.html#K5" title="POLYNOM7:func.5">*</a> <font color="Maroon" title="c16">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T17" title="POLYRED:th.17">POLYRED:17</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="polynom7.html#K4" title="POLYNOM7:func.4">|</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c16">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T27" title="POLYNOM7:th.27">POLYNOM7:27</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</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#K8" title="POLYNOM1:func.8">1_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">*'</a> <font color="Maroon" title="c16">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#T20" title="POLYNOM7:th.20">POLYNOM7:20</a></span>
<br/>.= 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T30" title="POLYNOM1:th.30">POLYNOM1:30</a></span>
;<br/>
<a NAME="E54:42"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:42"><span class="lab"><font color="Green" title="E52">A32</font></span></a>, <a class="txt" href="#E49:42"><span class="lab"><font color="Green" title="E55">A35</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/>
<a NAME="E55:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E58">A38</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> implies <font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:42_15_1_1"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><a NAME="E2:42_15_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">1bag</font>
 <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/><span class="kw">hence </span><a NAME="E3:42_15_1_1"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E2:42_15_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E3:42_15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A39</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/><span class="lab"><font color="Green" title="E60">A40</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T15" title="POLYNOM1:th.15">POLYNOM1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c16">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c5">1bag</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T17" title="POLYNOM1:th.17">POLYNOM1:17</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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="#E28:42"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="funct_7.html#T31" title="FUNCT_7:th.31">FUNCT_7:31</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#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
;<br/><a NAME="E5:42_15_1"/>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></span>;<br/><a NAME="E6:42_15_1"/><span class="kw">then </span>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c13">q</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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T20" title="POLYNOM1:th.20">POLYNOM1:20</a></span>;<br/><a NAME="E7:42_15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A41</font></span>: 
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c5">1bag</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="groeb_1.html#T5" target="_self" title="GROEB_1:th.5">Th5</a></span>;<br/><span class="kw">assume </span><a NAME="E8:42_15_1"/><span class="lab"><font color="Green" title="E62">A42</font></span>: 
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E9:42_15_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">u</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><a NAME="E10:42_15_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_15_1"><span class="lab"><font color="Green" title="E62">A42</font></span></a>, <a class="txt" href="#E7:42_15_1"><span class="lab"><font color="Green" title="E61">A41</font></span></a>, <a class="txt" href="#E3:42_15_1"><span class="lab"><font color="Green" title="E59">A39</font></span></a>, <a class="txt" href="#E4:42_15_1"><span class="lab"><font color="Green" title="E60">A40</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E11:42_15_1"/>
<font color="Maroon" title="c22">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI: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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:42_15"/>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>) is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:42_15"/><span class="kw">then </span>
 <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>),<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:42"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="txt" href="#E52:42"><span class="lab"><font color="Green" title="E56">A36</font></span></a>, <a class="ref" href="rewrite1.html#D24" title="REWRITE1:def.24">REWRITE1:def 24</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:42_15"/><span class="lab"><font color="Green" title="E59">A43</font></span>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>),<font color="Maroon" title="c22">c</font>
 <span class="kw">and </span><br/><a NAME="E6:42_15"/><span class="lab"><font color="Green" title="E60">A44</font></span>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>) <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font>,<font color="Maroon" title="c22">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" title="REWRITE1:def.7">REWRITE1:def 7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">c</font> = <font color="Maroon" title="c22">c</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="txt" href="#E5:42_15"><span class="lab"><font color="Green" title="E59">A43</font></span></a>, <a class="txt" href="groeb_1.html#E20"><span class="lab"><font color="Green" title="E13">Lm4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E9:42_15"/><span class="lab"><font color="Green" title="E61">A45</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E10:42_15"/><span class="lab"><font color="Green" title="E62">A46</font></span>: 
<font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 <span class="kw">and </span><br/><a NAME="E11:42_15"/><span class="lab"><font color="Green" title="E63">A47</font></span>: 
<font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">c</font>
 <span class="kw">and </span><br/><a NAME="E12:42_15"/><span class="lab"><font color="Green" title="E64">A48</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">s</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">s</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_15"><span class="lab"><font color="Green" title="E59">A43</font></span></a>, <a class="ref" href="rewrite1.html#T11" title="REWRITE1:th.11">REWRITE1:11</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:42_15_2"/><span class="lab"><font color="Green" title="E65">A49</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_15"><span class="lab"><font color="Green" title="E61">A45</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E2:42_15_2"/><span class="lab"><font color="Green" title="E66">A50</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E3:42_15_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:42_15_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_15_2"><span class="lab"><font color="Green" title="E65">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:42_15_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:42_15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A51</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_15_2"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:42_15_2"/><span class="lab"><font color="Green" title="E68">A52</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_15_2"><span class="lab"><font color="Green" title="E65">A49</font></span></a>, <a class="txt" href="#E2:42_15_2"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">f9</font>, <font color="Maroon" title="c26">h9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:42_15_2"/><span class="lab"><font color="Green" title="E69">A53</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">f9</font>,<font color="Maroon" title="c26">h9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:42_15_2"/>
<font color="Maroon" title="c25">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:42_15_2"/><span class="lab"><font color="Green" title="E70">A54</font></span>: 
<font color="Maroon" title="c26">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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="txt" href="#E10:42_15"><span class="lab"><font color="Green" title="E62">A46</font></span></a>, <a class="txt" href="#E12:42_15"><span class="lab"><font color="Green" title="E64">A48</font></span></a>, <a class="txt" href="#E6:42_15_2"><span class="lab"><font color="Green" title="E67">A51</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/><font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">f9</font>,<font color="Maroon" title="c26">h9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_15_2"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c26">h9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">c9</font> = <font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 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="txt" href="#E11:42_15_2"><span class="lab"><font color="Green" title="E70">A54</font></span></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/><a NAME="E14:42_15_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_15"><span class="lab"><font color="Green" title="E64">A48</font></span></a>, <a class="txt" href="#E7:42_15_2"><span class="lab"><font color="Green" title="E68">A52</font></span></a>, <a class="txt" href="#E6:42_15_2"><span class="lab"><font color="Green" title="E67">A51</font></span></a></span>;<br/><a NAME="E15:42_15_2"/><span class="kw">then </span>
 <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>) <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c27">c9</font>,<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_15"><span class="lab"><font color="Green" title="E62">A46</font></span></a>, <a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">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="E17:42_15_2"/>
<font color="Maroon" title="c28">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">P</font>
 <span class="kw">and </span><br/><a NAME="E18:42_15_2"/><span class="lab"><font color="Green" title="E71">A55</font></span>: 
 <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>) <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c27">c9</font>,<font color="Maroon" title="c28">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E19:42_15_2"/>
 <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>) <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c28">g</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:42_15_2"><span class="lab"><font color="Green" title="E71">A55</font></span></a>, <a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E20:42_15_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T37" title="POLYRED:th.37">POLYRED:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E15:42_15"/><span class="lab"><font color="Green" title="E65">A56</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E16:42_15"/><span class="lab"><font color="Green" title="E66">A57</font></span>: 
<font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font>
 <span class="kw">and </span><br/><a NAME="E17:42_15"/><span class="lab"><font color="Green" title="E67">A58</font></span>: 
<font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">0_</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)
 <span class="kw">and </span><br/><a NAME="E18:42_15"/><span class="lab"><font color="Green" title="E68">A59</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <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>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">s</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">s</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_15"><span class="lab"><font color="Green" title="E60">A44</font></span></a>, <a class="txt" href="#E10:42_15"><span class="lab"><font color="Green" title="E62">A46</font></span></a>, <a class="txt" href="#E11:42_15"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="ref" href="rewrite1.html#T11" title="REWRITE1:th.11">REWRITE1:11</a></span>;<br/><div><span class="lab"><font color="Green" title="E69">A60</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_15_3"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_15_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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#T12" title="RLVECT_1:th.12">RLVECT_1:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_15_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T17" title="RLVECT_1:th.17">RLVECT_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:42_15_4"/><span class="lab"><font color="Green" title="E70">A61</font></span>: 
<font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">u9</font> = <font color="Maroon" title="c26">u</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> <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/><span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c27">u9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c27">u9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c27">u9</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c27">u9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T15" title="POLYNOM1:th.15">POLYNOM1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c27">u9</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c16">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c27">u9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T17" title="POLYNOM1:th.17">POLYNOM1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="c16">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c27">u9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E1:42_15_4"><span class="lab"><font color="Green" title="E70">A61</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">L</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="#E31:42"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="txt" href="#E1:42_15_4"><span class="lab"><font color="Green" title="E70">A61</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_1.html#D2" title="ALGSTR_1:def.2">ALGSTR_1:def 2</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:42_15_4"/>
<font color="Maroon" title="c26">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:42_15"><span class="lab"><font color="Green" title="E69">A60</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:42_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A62</font></span>: 
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_15"><span class="lab"><font color="Green" title="E58">A38</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><div><span class="lab"><font color="Green" title="E71">A63</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_15_5"/>
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="polyred.html#R7" title="POLYRED:pred.7">is_reducible_wrt</a> <font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">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="E3:42_15_5"/><span class="lab"><font color="Green" title="E72">A64</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c26">g</font>,<font color="Maroon" title="c20">P</font>,<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">consider </span><font color="Maroon" title="c27">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="E5:42_15_5"/><span class="lab"><font color="Green" title="E73">A65</font></span>: 
<font color="Maroon" title="c27">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">P</font>
 <span class="kw">and </span><br/><a NAME="E6:42_15_5"/><span class="lab"><font color="Green" title="E74">A66</font></span>: 
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c26">g</font>,<font color="Maroon" title="c27">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_15_5"><span class="lab"><font color="Green" title="E72">A64</font></span></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></span>;<br/><a NAME="E7:42_15_5"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st <font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c26">g</font>,<font color="Maroon" title="c27">h</font>,<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_15_5"><span class="lab"><font color="Green" title="E74">A66</font></span></a>, <a class="ref" href="polyred.html#D6" title="POLYRED:def.6">POLYRED:def 6</a></span>;<br/><a NAME="E8:42_15_5"/><span class="kw">then </span>
<font color="Maroon" title="c27">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K7" title="POLYNOM1:func.7">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="polyred.html#D5" title="POLYRED:def.5">POLYRED:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">h</font> = <font color="Maroon" title="c27">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="ref" href="polynom7.html#D1" title="POLYNOM7:def.1">POLYNOM7:def 1</a></span>;<br/><a NAME="E10:42_15_5"/>
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="polyred.html#R6" title="POLYRED:pred.6">is_reducible_wrt</a> <font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_15_5"><span class="lab"><font color="Green" title="E74">A66</font></span></a>, <a class="ref" href="polyred.html#D8" title="POLYRED:def.8">POLYRED:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">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="E12:42_15_5"/><span class="lab"><font color="Green" title="E75">A67</font></span>: 
<font color="Maroon" title="c29">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:42_15_5"/><span class="lab"><font color="Green" title="E76">A68</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>) <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c29">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T36" title="POLYRED:th.36">POLYRED:36</a></span>;<br/><a NAME="E14:42_15_5"/><span class="lab"><font color="Green" title="E77">A69</font></span>: 
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">1bag</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_5_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_5_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_15_5_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="E1:42_15_5_1_1_1"/>
( <font color="Maroon" title="c28">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font> or <font color="Maroon" title="c28">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_15_5"><span class="lab"><font color="Green" title="E73">A65</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_15_5_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:42_15_5_1_1_1_1"/>
<font color="Maroon" title="c28">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">1bag</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_15_5_1_1_1_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">1bag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="txt" href="#E39:42"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E37:42"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_15_5_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:42_15_5_1_1_1_2"/>
<font color="Maroon" title="c28">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">1bag</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_15_5_1_1_1_2"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">1bag</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E23:42"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:42_15_5_1"/>
 <a href="termord.html#K3" title="TERMORD:func.3">HT</a> (<font color="Maroon" title="c28">h</font>,<font color="Maroon" title="c2">T</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">1bag</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:42_15_5"/>
<font color="Maroon" title="c29">b9</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:42_15"><span class="lab"><font color="Green" title="E70">A62</font></span></a>, <a class="txt" href="#E12:42_15_5"><span class="lab"><font color="Green" title="E75">A67</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E16:42_15_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="#E13:42_15_5"><span class="lab"><font color="Green" title="E76">A68</font></span></a>, <a class="txt" href="#E14:42_15_5"><span class="lab"><font color="Green" title="E77">A69</font></span></a>, <a class="ref" href="pre_poly.html#T58" title="PRE_POLY:th.58">PRE_POLY:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_15_6"><span class="kw">now </span></a><div class="add"><a NAME="E1:42_15_6"/><span class="lab"><font color="Green" title="E72">A70</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_15"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E2:42_15_6"/><span class="lab"><font color="Green" title="E73">A71</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E3:42_15_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:42_15_6"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_15_6"><span class="lab"><font color="Green" title="E72">A70</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:42_15_6"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:42_15_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A72</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_15_6"><span class="lab"><font color="Green" title="E73">A71</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:42_15_6"/><span class="lab"><font color="Green" title="E75">A73</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_15_6"><span class="lab"><font color="Green" title="E72">A70</font></span></a>, <a class="txt" href="#E2:42_15_6"><span class="lab"><font color="Green" title="E73">A71</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">f9</font>, <font color="Maroon" title="c27">h9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:42_15_6"/><span class="lab"><font color="Green" title="E76">A74</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">f9</font>,<font color="Maroon" title="c27">h9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:42_15_6"/>
<font color="Maroon" title="c26">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K8" title="STRUCT_0:func.8">NonZero</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:42_15_6"/><span class="lab"><font color="Green" title="E77">A75</font></span>: 
<font color="Maroon" title="c27">h9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K11" title="POLYNOM1:func.11">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="txt" href="#E16:42_15"><span class="lab"><font color="Green" title="E66">A57</font></span></a>, <a class="txt" href="#E18:42_15"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E6:42_15_6"><span class="lab"><font color="Green" title="E74">A72</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/><font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">f9</font>,<font color="Maroon" title="c27">h9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_15_6"><span class="lab"><font color="Green" title="E76">A74</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c27">h9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">c9</font> = <font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 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="txt" href="#E11:42_15_6"><span class="lab"><font color="Green" title="E77">A75</font></span></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>;<br/><a NAME="E14:42_15_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> (<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42_15"><span class="lab"><font color="Green" title="E66">A57</font></span></a>, <a class="txt" href="#E18:42_15"><span class="lab"><font color="Green" title="E68">A59</font></span></a>, <a class="txt" href="#E7:42_15_6"><span class="lab"><font color="Green" title="E75">A73</font></span></a>, <a class="txt" href="#E6:42_15_6"><span class="lab"><font color="Green" title="E74">A72</font></span></a></span>;<br/><a NAME="E15:42_15_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">q</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c16">p</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c28">c9</font>,<font color="Maroon" title="c20">P</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D13" title="POLYRED:def.13">POLYRED:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E16:42_15_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:42_15"><span class="lab"><font color="Green" title="E71">A63</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E24:42_15"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:42"><span class="lab"><font color="Green" title="E54">A34</font></span></a>, <a class="txt" href="#E16:42_15"><span class="lab"><font color="Green" title="E66">A57</font></span></a>, <a class="txt" href="#E17:42_15"><span class="lab"><font color="Green" title="E67">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E57:42"/>
 not <font color="Maroon" title="c20">P</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groeb_1.html#D3" target="_self" title="GROEB_1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
