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<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="group_1.html#V5" title="GROUP_1:attr.5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">b</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> &amp; <font color="Olive" title="b4">d</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> &amp; <font color="Olive" title="b2">b</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">d</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Olive" title="b3">c</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Olive" title="b4">d</font></span>)</span></span><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>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">b</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> &amp; <font color="Maroon" title="c5">d</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> &amp; <font color="Maroon" title="c3">b</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">d</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">c</font> implies <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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span> <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#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c3">b</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> &amp; <font color="Maroon" title="c5">d</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> &amp; <font color="Maroon" title="c3">b</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">d</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span> <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#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c2">a</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c3">b</font>;<br/>
<b>set </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font>;<br/>
<b>set </b><font color="Maroon" title="c8">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span>;<br/>
<a NAME="E2:26"/><i><font color="Green" title="E19">A2</font></i>: 
<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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:26"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green" title="E20">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:26"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</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#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:26"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="gcd_1.html#T3" target="_self" title="GCD_1:th.3">Th3</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</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>
 
<b>by </b><i><a class="txt" href="gcd_1.html#E1:26"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></i>;<br/>
<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#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span> = 
<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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span>
<b>by </b><i><a class="txt" href="gcd_1.html#E2:26"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="gcd_1.html#E3:26"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="gcd_1.html#E4:26"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="gcd_1.html#D4" target="_self" title="GCD_1:def.4">Def4</a></i>
<br/>.= 
<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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span>
<b>by </b><i><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="E7:26"/>
<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="c3">b</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <font color="Maroon" title="c5">d</font></span>)</span> <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#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="gcd_1.html#K1" title="GCD_1:func.1">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c5">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="gcd_1.html#E5:26"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="gcd_1.html#T1" target="_self" title="GCD_1:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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