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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="group_1.html#V4" title="GROUP_1:attr.4">associative</a>  <a href="vectsp_1.html#L1" title="VECTSP_1:struct.1">multLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><a NAME="E1:17"/>
<font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">D</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:17"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/>
<a NAME="E4:17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E3:17"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon" title="c4">C</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c4">C</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">Def1</a></i>;<br/>


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