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<b>let </b><font color="Maroon">R</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="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#V8">left_unital</a> <a href="algstr_1.html#V1">left_zeroed</a> <a href="binom.html#V1">add-cancelable</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1" target="_self">add-closed</a> <a href="ideal_1.html#V2" target="_self">left-ideal</a> <a href="ideal_1.html#V3" target="_self">right-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ideal_1.html#V1" target="_self">add-closed</a> <a href="ideal_1.html#V2" target="_self">left-ideal</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:136"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">I</font>,<font color="Maroon">J</font> <a href="ideal_1.html#R4" target="_self">are_co-prime</a> 
 ;<br/>

<a NAME="E2:136"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">J</font></span>)</span> <a href="ideal_1.html#K15" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K15" target="_self">*'</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E3:136"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K12" target="_self">+</a> <font color="Maroon">J</font></span>)</span> <a href="ideal_1.html#K15" target="_self">*'</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ideal_1.html#D5" target="_self">Def5</a>, </i>;<br/>
<a NAME="E4:136"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K15" target="_self">*'</a> <font color="Maroon">J</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:136"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K15" target="_self">*'</a> <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:136"/>
<font color="Maroon">I</font> <a href="ideal_1.html#K15" target="_self">*'</a> <font color="Maroon">J</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K13" target="_self">/\</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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