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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>, <font color="Maroon">c<sub>5</sub></font> be   <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>let </b><font color="Maroon">c<sub>6</sub></font> be  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/>









<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:33"/>
<font color="Maroon">c<sub>6</sub></font> <a href="groeb_1.html#R1" target="_self">is_Groebner_basis_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:33"/><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#V9">locally-confluent</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:33"/><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#V7">confluent</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green">E25</font></i>: 
 <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/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="polyred.html#R11">are_congruent_mod</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/><a NAME="E2:33_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</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="polyred.html#T58">POLYRED:58</a></i>;<br/><b>hence </b><a NAME="E3:33_1"/>
 <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rewrite1.html#T56">REWRITE1:56</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_2"/><i><font color="Green">E27</font></i>: 
 <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:33_2"/>
(  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rewrite1.html#R4">is_a_normal_form_of</a> <font color="Maroon">c<sub>4</sub></font>, <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rewrite1.html#R4">is_a_normal_form_of</a> <font color="Maroon">c<sub>5</sub></font>, <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E4:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rewrite1.html#T55">REWRITE1:55</a></i>;<br/><a NAME="E3:33_2"/><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> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">c<sub>4</sub></font>, <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <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="Maroon">c<sub>5</sub></font>, <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D6">REWRITE1:def 6</a></i>;<br/><a NAME="E4:33_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font>, <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</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="rewrite1.html#T26">REWRITE1:26</a></i>;<br/><a NAME="E5:33_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="rewrite1.html#R2">are_convertible_wrt</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:33_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="rewrite1.html#T31">REWRITE1:31</a></i>;<br/><b>hence </b><a NAME="E6:33_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="polyred.html#R11">are_congruent_mod</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="polyred.html#T57">POLYRED:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:33"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="polyred.html#R11">are_congruent_mod</a> <font color="Maroon">c<sub>6</sub></font> <a href="ideal_1.html#K7">-Ideal</a>  iff  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rewrite1.html#K2">nf</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E5:33"><i><font color="Green">E26</font></i></a></i>;<br/>


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