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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">G</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font></span>)</span>;<br/>









<b>set </b><font color="Maroon" title="c7">R</font> =  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font>;<br/>
<b>assume </b><a NAME="E1:33"/>
<font color="Maroon" title="c6">G</font> <a href="groeb_1.html#R1" title="GROEB_1:pred.1">is_Groebner_basis_wrt</a> <font color="Maroon" title="c2">T</font>
 ;<br/>

<a NAME="E2:33"/><b>then </b>
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#D3" target="_self" title="GROEB_1:def.3">Def3</a></i>;<br/>
<a NAME="E3:33"/><b>then </b><i><font color="Green" title="E25">A1</font></i>: 
 <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> is <a href="rewrite1.html#V4" title="REWRITE1:attr.4">with_UN_property</a>
 
<b>by </b><i><a class="ref" href="groeb_1.html#T12" target="_self" title="GROEB_1:th.12">Th12</a>, <a class="ref" href="groeb_1.html#T13" target="_self" title="GROEB_1:th.13">Th13</a></i>;<br/>
<div><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"/><i><font color="Green" title="E26">A2</font></i>: 
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span>
 ;<br/><a NAME="E2:33_1"/>
(  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c4">p</font>, <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> &amp;  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c5">q</font>, <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rewrite1.html#T55" title="REWRITE1:th.55">REWRITE1:55</a></i>;<br/><a NAME="E3:33_1"/><b>then </b>
(  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">p</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> &amp;  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c5">q</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></i>;<br/><a NAME="E4:33_1"/><b>then </b>
( <font color="Maroon" title="c4">p</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> &amp;  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span>,<font color="Maroon" title="c5">q</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  <a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#T26" title="REWRITE1:th.26">REWRITE1:26</a></i>;<br/><b>hence </b><a NAME="E5:33_1"/>
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c6">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="groeb_1.html#E1:33_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="polyred.html#T57" title="POLYRED:th.57">POLYRED:57</a>, <a class="ref" href="rewrite1.html#T31" title="REWRITE1:th.31">REWRITE1:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33"/>
( <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="polyred.html#R11" title="POLYRED:pred.11">are_congruent_mod</a> <font color="Maroon" title="c6">G</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  iff  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="polyred.html#K3" title="POLYRED:func.3">PolyRedRel</a> <font color="Maroon" title="c6">G</font>,<font color="Maroon" title="c2">T</font></span>)</span> )
 <b>by </b><i><a class="txt" href="groeb_1.html#E3:33"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="polyred.html#T58" title="POLYRED:th.58">POLYRED:58</a>, <a class="ref" href="rewrite1.html#T56" title="REWRITE1:th.56">REWRITE1:56</a></i>;<br/>


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