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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</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#V2">admissible</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  non <a href="struct_0.html#V3">empty</a> <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="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1">add-closed</a> <a href="ideal_1.html#V2">left-ideal</a> <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  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <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>assume </b><a NAME="E1:60"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="groeb_1.html#R4" target="_self">is_reduced_wrt</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="groeb_1.html#R2" target="_self">is_Groebner_basis_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="groeb_1.html#R4" target="_self">is_reduced_wrt</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:60"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E3:60"/><i><font color="Green">E44</font></i>: 
(  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V9">locally-confluent</a> &amp;  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V9">locally-confluent</a> )
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="ref" href="groeb_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:60"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="ref" href="groeb_1.html#D4" target="_self">Def4</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="groeb_1.html#E1:60_1"><i><font color="Green">E47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1"><b>case </b></a><a NAME="E1:60_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:60_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="groeb_1.html#R3" target="_self">is_monic_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E1:60_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E5:60_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="groeb_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E6:60_1_1_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/><b>hence </b><a NAME="E7:60_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:60_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="termord.html#T17">TERMORD:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_2"><b>case </b></a><a NAME="E1:60_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:60_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="groeb_1.html#R3" target="_self">is_monic_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E1:60_1_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E5:60_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="groeb_1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E6:60_1_1_1_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/><b>hence </b><a NAME="E7:60_1_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:60_1_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="termord.html#T17">TERMORD:17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:60_1"/>
( <font color="Maroon">c<sub>7</sub></font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</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="E1:60_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/><a NAME="E2:60_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="groeb_1.html#R3" target="_self">is_monic_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:60_2"/>
 <a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="groeb_1.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:60_3"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:60_3_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:60_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:60_3_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:60_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:60_3"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60"><i><font color="Green">E43</font></i></a>, <a class="txt" href="groeb_1.html#E3:60_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <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/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:60_4"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:60_4"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_4_1"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_4_1_1_1"/>
(  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="groeb_1.html#E1:60_4_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_4_1_1_1_1"><b>case </b></a><a NAME="E1:60_4_1_1_1_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_4_1_1_1_1"/>
( not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_4_1_1_1_2"><b>case </b></a><a NAME="E1:60_4_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_4_1_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:60_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_5"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_5_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="groeb_1.html#E1:60_5"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_5_1_1_1"><b>case </b></a><a NAME="E1:60_5_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_5_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:60_5_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_5_1_1_2"><b>case </b></a><a NAME="E1:60_5_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_5_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:60_5_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:60_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:60"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E9:60"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_6"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:60_6"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K13">/\</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:60_6"/><b>then </b><i><font color="Green">E52</font></i>: 
(  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_6_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 </span><b>by </b><i><a class="txt" href="groeb_1.html#E2:60_6"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_6_1_1_1"><b>case </b></a><a NAME="E1:60_6_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_6_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_6_1_1_2"><b>case </b></a><a NAME="E1:60_6_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_6_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_6"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:60_6"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E14:60"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="polyred.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="polyred.html#T31">POLYRED:31</a></i>;<br/>
<a NAME="E15:60"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V7">confluent</a>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E16:60"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V4">with_UN_property</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E17:60"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V8">with_Church-Rosser_property</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E18:60"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">b<sub>1</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E19:60"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E20:60"/><b>then </b><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E21:60"/>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V7">confluent</a>
 
<b>by </b><i><a class="txt" href="groeb_1.html#E3:60"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E22:60"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V4">with_UN_property</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E23:60"/><b>then </b>
 <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="rewrite1.html#V8">with_Church-Rosser_property</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E24:60"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/> <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">b<sub>1</sub></font>, <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E25:60"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E26:60"/><b>then </b><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="polyred.html#R10">is_top_reducible_wrt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T17" target="_self">Th17</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_7"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="groeb_1.html#E12:60"><i><font color="Green">E50</font></i></a>, <a class="txt" href="groeb_1.html#E14:60"><i><font color="Green">E51</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_1"><b>suppose </b></a><a NAME="E1:60_7_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_7_1"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:60_7_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D15">IDEAL_1:def 15</a></i>;<br/><a NAME="E4:60_7_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E5:60"><i><font color="Green">E46</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E6:60_7_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E4:60_7_1"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E7:60_7_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:60_7_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#R3">divides</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E20:60"><i><font color="Green">E52</font></i></a>, <a class="ref" href="groeb_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E10:60_7_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E9:60_7_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E12:60_7_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E12:60_7_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E14:60_7_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D15">IDEAL_1:def 15</a></i>;<br/><a NAME="E15:60_7_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60_7_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E16:60_7_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60_7_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E17:60_7_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E18:60_7_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:60_7_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="polyred.html#T36">POLYRED:36</a></i>;<br/><a NAME="E19:60_7_1"/><b>then </b><i><font color="Green">E63</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_7_1_1"/>
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:60_7_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E12:60_7_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:60_7_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#NR10" target="_self">is_irreducible_wrt</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E4:60_7_1_1"/>
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E15:60_7_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:60_7_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_7_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:60_7_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E19:60_7_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E8:60_7_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:60_7_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><b>hence </b><a NAME="E9:60_7_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_7_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/></div><b>end;</b></div><a NAME="E21:60_7_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E11:60"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E22:60_7_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:60"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E23:60_7_1"/><b>then </b>
not <font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R2">&lt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T29">POLYRED:29</a></i>;<br/><a NAME="E24:60_7_1"/><b>then </b>
not  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R2">&lt;</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T32">POLYRED:32</a></i>;<br/><a NAME="E25:60_7_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#T5">TERMORD:5</a></i>;<br/><a NAME="E26:60_7_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E18:60_7_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="groeb_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E27:60_7_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E25:60_7_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>13</sub></font>;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_7_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="polyred.html#T2">POLYRED:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="groeb_1.html#E1:60_7_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</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="polyred.html#T3">POLYRED:3</a></i>
;<br/><b>hence </b><a NAME="E3:60_7_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E15:60_7_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>13</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E30:60_7_1"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E28:60_7_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E31:60_7_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E32:60_7_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K2">max</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="groeb_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E33:60_7_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E27:60_7_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="termord.html#T12">TERMORD:12</a></i>;<br/><font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E27:60_7_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="polynom1.html#D22">POLYNOM1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E6:60"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_1"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E6:60"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_1"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/><a NAME="E35:60_7_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E31:60_7_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E36:60_7_1"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T79">POLYNOM1:79</a></i>;<br/><a NAME="E37:60_7_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>;<br/><a NAME="E38:60_7_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_1.html#T5" target="_self">Th5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</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>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</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/><a NAME="E41:60_7_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="groeb_1.html#E3:60_7_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="groeb_1.html#E14:60_7_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="ideal_1.html#T15">IDEAL_1:15</a></i>;<br/><a NAME="E42:60_7_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4"><i><font color="Green">Lemma3</font></i></a></i>;<br/><a NAME="E43:60_7_1"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E28:60_7_1"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E44:60_7_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_7_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_7_1_4_1"/>
(  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font> or  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> )
 </span><b>by </b><i><a class="txt" href="groeb_1.html#E31:60_7_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="groeb_1.html#E38:60_7_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_1_4_1_1"><b>case </b></a><a NAME="E1:60_7_1_4_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:60_7_1_4_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#R3">divides</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E26:60"><i><font color="Green">E53</font></i></a>, <a class="txt" href="groeb_1.html#E44:60_7_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="groeb_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:60_7_1_4_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E3:60_7_1_4_1_1"><i><font color="Green">E73</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:60_7_1_4_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_1_4_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:60_7_1_4_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60_7_1_4_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="groeb_1.html#E3:60_7_1_4_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_1_4_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="polyred.html#T36">POLYRED:36</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:60_7_1_4_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><a NAME="E11:60_7_1_4_1_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_7_1_4_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_1_4_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="termord.html#T10">TERMORD:10</a></i>;<br/><a NAME="E12:60_7_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E33:60_7_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="groeb_1.html#E35:60_7_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><a NAME="E13:60_7_1_4_1_1"/><b>then </b>
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:60_7_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_1_4_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:60_7_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:60_7_1_4_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E16:60_7_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/><b>hence </b><a NAME="E17:60_7_1_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_1_4_1_2"><b>case </b></a><a NAME="E1:60_7_1_4_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:60_7_1_4_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#R3">divides</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E20:60"><i><font color="Green">E52</font></i></a>, <a class="txt" href="groeb_1.html#E44:60_7_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="groeb_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:60_7_1_4_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E3:60_7_1_4_1_2"><i><font color="Green">E73</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:60_7_1_4_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_1_4_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:60_7_1_4_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60_7_1_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="groeb_1.html#E3:60_7_1_4_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_1_4_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="polyred.html#T36">POLYRED:36</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:60_7_1_4_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><a NAME="E11:60_7_1_4_1_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_7_1_4_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_1_4_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="termord.html#T10">TERMORD:10</a></i>;<br/><a NAME="E12:60_7_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E27:60_7_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="groeb_1.html#E33:60_7_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="groeb_1.html#E35:60_7_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><a NAME="E13:60_7_1_4_1_2"/><b>then </b>
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:60_7_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_1_4_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:60_7_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:60_7_1_4_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E16:60_7_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/><b>hence </b><a NAME="E17:60_7_1_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E46:60_7_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_2"><b>suppose </b></a><a NAME="E1:60_7_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:60_7_2"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:60_7_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D15">IDEAL_1:def 15</a></i>;<br/><a NAME="E4:60_7_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E5:60"><i><font color="Green">E46</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_2"><i><font color="Green">E54</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E6:60_7_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E4:60_7_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E7:60_7_2"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:60_7_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#R3">divides</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E26:60"><i><font color="Green">E53</font></i></a>, <a class="ref" href="groeb_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E10:60_7_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E9:60_7_2"><i><font color="Green">E57</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E12:60_7_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E14:60_7_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#D15">IDEAL_1:def 15</a></i>;<br/><a NAME="E15:60_7_2"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60_7_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E16:60_7_2"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E17:60_7_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E9:60_7_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="polyred.html#T36">POLYRED:36</a></i>;<br/><a NAME="E18:60_7_2"/><b>then </b><i><font color="Green">E61</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_7_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_7_2_1"/>
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:60_7_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:60_7_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#NR10" target="_self">is_irreducible_wrt</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E4:60_7_2_1"/>
not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_7_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:60_7_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_7_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:60_7_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E18:60_7_2"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E8:60_7_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E7:60_7_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><b>hence </b><a NAME="E9:60_7_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_7_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/></div><b>end;</b></div><a NAME="E20:60_7_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E11:60"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E21:60_7_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E14:60"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E22:60_7_2"/><b>then </b>
not <font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R2">&lt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T29">POLYRED:29</a></i>;<br/><a NAME="E23:60_7_2"/><b>then </b>
not  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R2">&lt;</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#T32">POLYRED:32</a></i>;<br/><a NAME="E24:60_7_2"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#T5">TERMORD:5</a></i>;<br/><a NAME="E25:60_7_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E17:60_7_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="groeb_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E26:60_7_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E24:60_7_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>13</sub></font>;<br/><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_7_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_7_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="polyred.html#T2">POLYRED:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="groeb_1.html#E1:60_7_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T80">POLYNOM1:80</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26">0_</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="polyred.html#T3">POLYRED:3</a></i>
;<br/><b>hence </b><a NAME="E3:60_7_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E2:60_7_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="polynom1.html#T82">POLYNOM1:82</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K25">-</a> <font color="Maroon">c<sub>13</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E29:60_7_2"/>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E27:60_7_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polynom7.html#T1">POLYNOM7:1</a></i>;<br/><a NAME="E30:60_7_2"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D6">TERMORD:def 6</a></i>;<br/><a NAME="E31:60_7_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K2">max</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="groeb_1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E32:60_7_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E26:60_7_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="termord.html#T12">TERMORD:12</a></i>;<br/><font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p1">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K15">.</a> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E26:60_7_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="polynom1.html#D22">POLYNOM1:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K15">.</a> <span class="p3">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="termord.html#D7">TERMORD:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="termord.html#K4">HC</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E6:60"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_2"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="groeb_1.html#E6:60"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/><a NAME="E34:60_7_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E30:60_7_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E35:60_7_2"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K23">+</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T79">POLYNOM1:79</a></i>;<br/><a NAME="E36:60_7_2"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="polynom1.html#K24">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D23">POLYNOM1:def 23</a></i>;<br/><a NAME="E37:60_7_2"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_1.html#T5" target="_self">Th5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</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>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</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/><a NAME="E40:60_7_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="groeb_1.html#E3:60_7_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="groeb_1.html#E14:60_7_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="ideal_1.html#T15">IDEAL_1:15</a></i>;<br/><a NAME="E41:60_7_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4"><i><font color="Green">Lemma3</font></i></a></i>;<br/><a NAME="E42:60_7_2"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E27:60_7_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E43:60_7_2"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_7_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_7_2_4_1"/>
(  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font> or  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font> )
 </span><b>by </b><i><a class="txt" href="groeb_1.html#E30:60_7_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="groeb_1.html#E37:60_7_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_2_4_1_1"><b>case </b></a><a NAME="E1:60_7_2_4_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:60_7_2_4_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#R3">divides</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E20:60"><i><font color="Green">E52</font></i></a>, <a class="txt" href="groeb_1.html#E43:60_7_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="groeb_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:60_7_2_4_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E3:60_7_2_4_1_1"><i><font color="Green">E71</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:60_7_2_4_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_2_4_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:60_7_2_4_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60_7_2_4_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="groeb_1.html#E3:60_7_2_4_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_2_4_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="polyred.html#T36">POLYRED:36</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:60_7_2_4_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><a NAME="E11:60_7_2_4_1_1"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_7_2_4_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_2_4_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="termord.html#T10">TERMORD:10</a></i>;<br/><a NAME="E12:60_7_2_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E32:60_7_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="groeb_1.html#E34:60_7_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><a NAME="E13:60_7_2_4_1_1"/><b>then </b>
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:60_7_2_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_2_4_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:60_7_2_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:60_7_2_4_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E16:60_7_2_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/><b>hence </b><a NAME="E17:60_7_2_4_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E2:60_7_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_7_2_4_1_2"><b>case </b></a><a NAME="E1:60_7_2_4_1_2"/><i><font color="Green">E70</font></i>: 
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:60_7_2_4_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="groeb_1.html#K2" target="_self">HT</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#R3">divides</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_1.html#E26:60"><i><font color="Green">E53</font></i></a>, <a class="txt" href="groeb_1.html#E43:60_7_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="groeb_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E4:60_7_2_4_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="termord.html#K3">HT</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> where B is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</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="txt" href="groeb_1.html#E3:60_7_2_4_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:60_7_2_4_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="pboole.html#R6">=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom1.html#K26">0_</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_2_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="polynom7.html#D2">POLYNOM7:def 2</a></i>;<br/><a NAME="E8:60_7_2_4_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R6">is_reducible_wrt</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60_7_2_4_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="groeb_1.html#E3:60_7_2_4_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_2_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="polyred.html#T36">POLYRED:36</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:60_7_2_4_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R4">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D8">POLYRED:def 8</a></i>;<br/><a NAME="E11:60_7_2_4_1_2"/>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="termord.html#R1">&lt;=</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:60_7_2_4_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="groeb_1.html#E6:60_7_2_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="termord.html#T10">TERMORD:10</a></i>;<br/><a NAME="E12:60_7_2_4_1_2"/><b>then </b>
 <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="termord.html#K3">HT</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E26:60_7_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="groeb_1.html#E32:60_7_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="groeb_1.html#E34:60_7_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="termord.html#T7">TERMORD:7</a></i>;<br/><a NAME="E13:60_7_2_4_1_2"/><b>then </b>
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E14:60_7_2_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E6:60_7_2_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:60_7_2_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R5">reduces_to</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E10:60_7_2_4_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="polyred.html#D7">POLYRED:def 7</a></i>;<br/><a NAME="E16:60_7_2_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="polyred.html#R7">is_reducible_wrt</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="groeb_1.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="groeb_1.html#K1" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="polyred.html#D9">POLYRED:def 9</a></i>;<br/><b>hence </b><a NAME="E17:60_7_2_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="groeb_1.html#E12:60_7_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="groeb_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E45:60_7_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
