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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#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#V7" title="STRUCT_0:attr.7">trivial</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> <br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font><br/> for <font color="Olive" title="b5">P</font>, <font color="Olive" title="b6">Q</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font></span>)</span>  st <font color="Olive" title="b5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b6">Q</font> &amp; <font color="Olive" title="b3">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b5">P</font>,<font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b3">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b6">Q</font>,<font color="Olive" title="b1">T</font></span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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> <br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b4">P</font>, <font color="Olive" title="b5">Q</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font></span>)</span>  st <font color="Olive" title="b4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">Q</font> &amp; <font color="Olive" title="b2">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b4">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b2">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b5">Q</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st <font color="Olive" title="b3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">Q</font> &amp; <font color="Olive" title="b1">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b1">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b4">Q</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>  st <font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Olive" title="b2">Q</font>,<font color="Maroon" title="c2">T</font></span><br/><b>let </b><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c2">T</font> )</span><br/>









<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font> or <font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c2">T</font> )</span><br/>

<b>assume </b><a NAME="E2:7"/>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c2">T</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">p</font> being   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><b> such that </b><br/><a NAME="E4:7"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>;<br/>
<b>thus </b><a NAME="E5:7"/>
<font color="Maroon" title="c4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c7">Q</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="groeb_1.html#E4:7"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="polyred.html#D7" title="POLYRED:def.7">POLYRED:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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