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<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">aa</font>, <font color="Olive" title="b2">bb</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>  st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">aa</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">bb</font> &amp; <font color="Olive" title="b1">aa</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Olive" title="b2">bb</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c4">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">aa</font>, <font color="Maroon" title="c6">bb</font> be   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">aa</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">bb</font> &amp; <font color="Maroon" title="c5">aa</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c6">bb</font> implies <font color="Maroon" title="c3">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c4">b</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:101"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">aa</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">bb</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">aa</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c6">bb</font> or <font color="Maroon" title="c3">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c4">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:101"/>
<font color="Maroon" title="c5">aa</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c6">bb</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E3:101"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A2</font></span>: 
( <font color="Maroon" title="c5">aa</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c6">bb</font> &amp; <font color="Maroon" title="c6">bb</font> <a href="gaussint.html#R1" title="GAUSSINT:pred.1">Divides</a> <font color="Maroon" title="c5">aa</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">cca</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><br/><a NAME="E5:101"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c6">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">aa</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">cca</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">ccb</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><br/><a NAME="E7:101"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c5">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">bb</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">ccb</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">ca</font> = <font color="Maroon" title="c7">cca</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T3" target="_self" onmouseover="rs('gaussint/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">cb</font> = <font color="Maroon" title="c8">ccb</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T3" target="_self" onmouseover="rs('gaussint/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E10:101"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">ca</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E5:101"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E11:101"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" onmouseover="rs('gcd_1/D1')" onmouseout="rh()">GCD_1:def 1</a></span>;<br/>
<a NAME="E12:101"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">cb</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E7:101"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E13:101"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="gcd_1.html#R2" title="GCD_1:pred.2">divides</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gcd_1.html#D1" onmouseover="rs('gcd_1/D1')" onmouseout="rh()">GCD_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:101"/>
<font color="Maroon" title="c3">a</font> <a href="gcd_1.html#R4" title="GCD_1:pred.4">is_associated_to</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:101"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="gcd_1.html#D3" onmouseover="rs('gcd_1/D3')" onmouseout="rh()">GCD_1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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