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<span class="kw">let </span><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:()"><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#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#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b3">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><br/>  for <font color="Olive" title="b4">f</font> being   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font><br/>  for <font color="Olive" title="b5">g</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font><br/>  for <font color="Olive" title="b6">f9</font>, <font color="Olive" title="b7">g9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f9</font> &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">g9</font> &amp; <font color="Olive" title="b4">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b5">g</font>,<font color="Olive" title="b3">P</font>,<font color="Olive" title="b1">T</font> holds <br/><font color="Olive" title="b6">f9</font>,<font color="Olive" title="b7">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Olive" title="b3">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </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#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#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#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b2">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><br/>  for <font color="Olive" title="b3">f</font> being   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b4">g</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b5">f9</font>, <font color="Olive" title="b6">g9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f9</font> &amp; <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">g9</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="b2">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b5">f9</font>,<font color="Olive" title="b6">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Olive" title="b2">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><span class="kw">let </span><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#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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><br/>  for <font color="Olive" title="b2">f</font> being   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><br/>  for <font color="Olive" title="b3">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><br/>  for <font color="Olive" title="b4">f9</font>, <font color="Olive" title="b5">g9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f9</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g9</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="b1">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b4">f9</font>,<font color="Olive" title="b5">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Olive" title="b1">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</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#K11" title="POLYNOM1:func.11">Polynom-Ring</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a>  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font><br/>  for <font color="Olive" title="b2">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><br/>  for <font color="Olive" title="b3">f9</font>, <font color="Olive" title="b4">g9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f9</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g9</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="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b3">f9</font>,<font color="Olive" title="b4">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <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>; <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">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><br/>  for <font color="Olive" title="b2">f9</font>, <font color="Olive" title="b3">g9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f9</font> &amp; <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g9</font> &amp; <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Olive" title="b1">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b2">f9</font>,<font color="Olive" title="b3">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f9</font>, <font color="Olive" title="b2">g9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>  st <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f9</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g9</font> &amp; <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> holds <br/><font color="Olive" title="b1">f9</font>,<font color="Olive" title="b2">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">f9</font>, <font color="Maroon" title="c8">g9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>; <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="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f9</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g9</font> &amp; <font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> implies <font color="Maroon" title="c7">f9</font>,<font color="Maroon" title="c8">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:93"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f9</font>
 <span class="kw">and </span><br/><a NAME="E2:93"/><span class="lab"><font color="Green" title="E71">A2</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g9</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="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c2">T</font> or <font color="Maroon" title="c7">f9</font>,<font color="Maroon" title="c8">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">R</font> =  <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>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> =  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c6">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#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>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#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/>
<font color="Maroon" title="c11">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">g9</font> = 
<span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E71">A2</font></span></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</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="polyred.html#T3" target="_self" title="POLYRED:th.3">Th3</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</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">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>
;<br/>
<a NAME="E6:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A3</font></span>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T6" title="RLVECT_1:th.6">RLVECT_1:6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:93"/>
<font color="Maroon" title="c5">f</font> <a href="polyred.html#R5" title="POLYRED:pred.5">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">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"> <font color="Maroon" title="c7">f9</font>,<font color="Maroon" title="c8">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p</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="E9:93"/><span class="lab"><font color="Green" title="E73">A4</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 <span class="kw">and </span><br/><a NAME="E10:93"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="polyred.html#R4" title="POLYRED:pred.4">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#D7" target="_self" title="POLYRED:def.7">Def7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">b</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="E12:93"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="polyred.html#R3" title="POLYRED:pred.3">reduces_to</a> <font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c2">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:93"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="polyred.html#D6" target="_self" title="POLYRED:def.6">Def6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">s</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="E14:93"/>
<font color="Maroon" title="c14">s</font> <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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">and </span><br/><a NAME="E15:93"/><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</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="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">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"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:93"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="polyred.html#D5" target="_self" title="POLYRED:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">P</font> = <font color="Maroon" title="c4">P</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#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="#E9:93"><span class="lab"><font color="Green" title="E73">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">q9</font> = <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">r</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><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>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</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/><a NAME="E1:93_2"/><span class="lab"><font color="Green" title="E77">A8</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></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"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</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="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/><span class="kw">assume </span><a NAME="E2:93_2"/>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</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="E3:93_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E77">A8</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="E4:93_2"/>
<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>
<a NAME="E18:93"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">r</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c20">p9</font> = <font color="Maroon" title="c12">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#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>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">m</font> =  <a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#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>;<br/><span class="kw">let </span><font color="Maroon" title="c22">i</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">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="c19">r</font> implies  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> ex <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c15">P</font> st <font color="Maroon" title="c19">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">v</font> )</span><br/><span class="kw">assume </span><a NAME="E3:93_3"/><span class="lab"><font color="Green" title="E77">A9</font></span>: 
<font color="Maroon" title="c22">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="c19">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> ex <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c15">P</font> st <font color="Maroon" title="c19">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">v</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c23">p9</font> = <font color="Maroon" title="c20">p9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#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">reconsider </span><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c21">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#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="lab"><font color="Green" title="E78">A10</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c23">p9</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<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>)</span> = 
<font color="Maroon" title="c24">m</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c23">p9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" title="VECTSP_1:def.4">VECTSP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>
;<br/><a NAME="E7:93_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><a NAME="E8:93_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_3"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:93_3"/><span class="kw">then </span>
<font color="Maroon" title="c19">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/><span class="kw">hence </span><a NAME="E10:93_3"/>
 ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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> ex <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c15">P</font> st <font color="Maroon" title="c19">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:93"><span class="lab"><font color="Green" title="E73">A4</font></span></a>, <a class="txt" href="#E3:93_3"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E6:93_3"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</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="c20">r</font> = <font color="Maroon" title="c19">r</font> as    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c15">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ideal_1.html#D8" title="IDEAL_1:def.8">IDEAL_1:def 8</a></span>;<br/>
<a NAME="E22:93"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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>;<br/>
<a NAME="E23:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A11</font></span>: 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c20">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binom.html#T3" title="BINOM:th.3">BINOM:3</a></span>;<br/>
<a NAME="E24:93"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K1" title="POLYNOM7:func.1">Monom</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">p</font>,<font color="Maroon" title="c2">T</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c14">s</font>)</span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">*'</a> <font color="Maroon" title="c12">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T22" target="_self" title="POLYRED:th.22">Th22</a></span>;<br/>
<a NAME="E25:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:93"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="ideal_1.html#T60" title="IDEAL_1:th.60">IDEAL_1:60</a></span>;<br/>
<span class="lab"><font color="Green" title="E79">A13</font></span>: <font color="Maroon" title="c5">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c6">g</font> = 
<font color="Maroon" title="c5">f</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c6">g</font></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/>.= 
<font color="Maroon" title="c7">f9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c8">g9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E70">A1</font></span></a>, <a class="txt" href="#E6:93"><span class="lab"><font color="Green" title="E72">A3</font></span></a>, <a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></span>
<br/>.= 
<font color="Maroon" title="c7">f9</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c8">g9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
;<br/>
<a NAME="E27:93"/><span class="lab"><font color="Green" title="E80">a1</font></span>: 
 <a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p4">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p3">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T19" title="POLYNOM1:th.19">POLYNOM1:19</a></span>;<br/>
<font color="Maroon" title="c5">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <font color="Maroon" title="c6">g</font> = 
<font color="Maroon" title="c5">f</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p5">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p4">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:93"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p0">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p5">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span></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/>.= 
<font color="Maroon" title="c5">f</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</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="c5">f</font></span>)</span> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p3">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p0">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p5">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T1" target="_self" title="POLYRED:th.1">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">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"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:93"><span class="lab"><font color="Green" title="E80">a1</font></span></a>, <a class="ref" href="polynom1.html#T21" title="POLYNOM1:th.21">POLYNOM1:21</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#K5" title="POLYNOM1:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</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="c12">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"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T3" target="_self" title="POLYRED:th.3">Th3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p2">(<span class="default"><a href="termord.html#K4" title="TERMORD:func.4">HC</a> (<font color="Maroon" title="c12">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="p1">(<span class="default"><font color="Maroon" title="c14">s</font> <a href="polyred.html#K1" title="POLYRED:func.1">*'</a> <font color="Maroon" title="c12">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyred.html#T2" target="_self" title="POLYRED:th.2">Th2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E29:93"/>
<font color="Maroon" title="c7">f9</font>,<font color="Maroon" title="c8">g9</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c4">P</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:93"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="txt" href="#E26:93"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="ref" href="polyred.html#D14" target="_self" title="POLYRED:def.14">Def14</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/>


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