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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V7" title="ALGSTR_0:attr.7">add-cancelable</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="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#V5" title="VECTSP_1:attr.5">distributive</a> <a href="vectsp_1.html#V6" title="VECTSP_1:attr.6">left_unital</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V2" title="IDEAL_1:attr.2">left-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/>





<b>assume </b><a NAME="E1:135"/><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c2">I</font>,<font color="Maroon" title="c3">J</font> <a href="ideal_1.html#R4" title="IDEAL_1:pred.4">are_co-prime</a> 
 ;<br/>

<a NAME="E2:135"/><i><font color="Green" title="E59">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="ideal_1.html#K15" title="IDEAL_1:func.15">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K15" title="IDEAL_1:func.15">*'</a> <font color="Maroon" title="c3">J</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T84" target="_self" title="IDEAL_1:th.84">Th84</a></i>;<br/>
<a NAME="E3:135"/>
<font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K12" title="IDEAL_1:func.12">+</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="ideal_1.html#K15" title="IDEAL_1:func.15">*'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c3">J</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E1:135"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="ideal_1.html#T86" target="_self" title="IDEAL_1:th.86">Th86</a></i>;<br/>
<a NAME="E4:135"/><b>then </b><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K15" title="IDEAL_1:func.15">*'</a> <font color="Maroon" title="c3">J</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E2:135"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:135"/>
<font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K15" title="IDEAL_1:func.15">*'</a> <font color="Maroon" title="c3">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c3">J</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T82" target="_self" title="IDEAL_1:th.82">Th82</a></i>;<br/>
<b>hence </b><a NAME="E6:135"/>
<font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K15" title="IDEAL_1:func.15">*'</a> <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K13" title="IDEAL_1:func.13">/\</a> <font color="Maroon" title="c3">J</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:135"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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