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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:31"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>, <font color="Maroon">c<sub>15</sub></font> =  <a href="polynom7.html#K1">Monom</a> <span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<i><font color="Green">E20</font></i>:  <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom7.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="polyred.html#K1">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K25">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p3">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polyred.html#T22">POLYRED:22</a></i>
;<br/>
<a NAME="E7:31"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  )
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:31"><i><font color="Green">E19</font></i></a>, <a class="txt" href="groeb_2.html#E27"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<a NAME="E8:31"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Maroon">c<sub>17</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  )
 
<b>by </b><i><a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a></i>;<br/>
<a NAME="E9:31"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#T16">IDEAL_1:16</a></i>;<br/>
<a NAME="E10:31"/>
( <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><a href="polynom7.html#K1">Monom</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="groeb_2.html#K2" target="_self">lcm</a> <span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="groeb_2.html#K1" target="_self">/</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="polynom1.html#K29">*'</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<b>hence </b><a NAME="E11:31"/>
 <a href="groeb_2.html#K3" target="_self">S-Poly</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="groeb_2.html#E6:31"><i><font color="Green">E20</font></i></a>, <a class="txt" href="groeb_2.html#E9:31"><i><font color="Green">E21</font></i></a>, <a class="txt" href="groeb_2.html#E28"><i><font color="Green">Lemma17</font></i></a></i>;<br/>


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