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<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>thus </b><a NAME="E1:18"/>
<font color="Maroon" title="c2">a</font> <a href="gcd_1.html#R1" title="GCD_1:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">b</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">GCD_1:def 1</a><br/>

<b>thus </b><a NAME="E1:18_1"/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a>
 ;<br/>

<b>take </b>
<font color="Maroon" title="c2">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="gcd_1.html#D1" target="_self" title="GCD_1:def.1">GCD_1:def 1</a><br/>

<b>thus </b><a NAME="E3:18"/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:18"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a></i>;<br/>


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