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<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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="E1:23"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/>

<div><i><font color="Green" title="E17">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> )
 ;<br/><b>set </b><font color="Maroon" title="c5">A1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font>;<br/><a NAME="E2:23_1"/><i><font color="Green" title="E19">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="gcd_1.html#E1:23_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/><b>set </b><font color="Maroon" title="c6">A2</font> = <font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font>;<br/><a NAME="E3:23_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="gcd_1.html#E1:23_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/><a NAME="E4:23_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:23_1"><i><font color="Green" title="E19">A4</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="E5:23_1"/>
( <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="gcd_1.html#T1" target="_self" title="GCD_1:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_2"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">B</font> )
 ;<br/><b>set </b><font color="Maroon" title="c5">A1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font>;<br/><a NAME="E2:23_2"/><i><font color="Green" title="E19">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="gcd_1.html#E1:23_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/><b>set </b><font color="Maroon" title="c6">A2</font> = <font color="Maroon" title="c3">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font>;<br/><a NAME="E3:23_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="gcd_1.html#E1:23_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/><a NAME="E4:23_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:23_2"><i><font color="Green" title="E19">A6</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="E5:23_2"/>
( <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gcd_1.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="gcd_1.html#T1" target="_self" title="GCD_1:th.1">Th1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:23"/>
( ( <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> ) &amp; ( <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">C</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c4">C</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="gcd_1.html#E2:23"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>


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