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<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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">x</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font> iff  ex <font color="Olive" title="b1">c</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> st <br/>( <font color="Olive" title="b1">c</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font> ) )</span><br/>

<div><span class="kw">hereby </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"> (  ex <font color="Olive" title="b1">c</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> st <br/>( <font color="Olive" title="b1">c</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font> ) implies <font color="Maroon" title="c3">x</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:104_1"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</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">  ex <font color="Olive" title="b1">c</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> st <br/>( <font color="Olive" title="b1">c</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</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">consider </span><font color="Maroon" title="c7">c1</font> being   <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"> such that </span><br/><a NAME="E4:104_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
( <font color="Maroon" title="c7">c1</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c6">y1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:104_1"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="gaussint.html#T38" target="_self" onmouseover="rs('gaussint/T38')" onmouseout="rh()">Th38</a>, <a class="ref" href="gcd_1.html#T18" onmouseover="rs('gcd_1/T18')" onmouseout="rh()">GCD_1:18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">c1</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E6:104_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c8">c</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="gaussint.html#T39" target="_self" onmouseover="rs('gaussint/T39')" onmouseout="rh()">Th39</a></span>;<br/><a NAME="E7:104_1"/>
<font color="Maroon" title="c7">c1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E8:104_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> st <br/>( <font color="Olive" title="b1">c</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:104_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="txt" href="#E6:104_1"><span class="lab"><font color="Green" title="E46">A3</font></span></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/></div>
<span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c5">c0</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><a NAME="E3:104"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
( <font color="Maroon" title="c5">c0</font> is  <a href="gaussint.html#V3" title="GAUSSINT:attr.3">g_int_unit</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</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">x</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">xx</font> = <font color="Maroon" title="c3">x</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="c7">yy</font> = <font color="Maroon" title="c4">y</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="c8">c</font> = <font color="Maroon" title="c5">c0</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="E7:104"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c8">c</font> is  <a href="gcd_1.html#V1" title="GCD_1:attr.1">unital</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="gaussint.html#T39" target="_self" onmouseover="rs('gaussint/T39')" onmouseout="rh()">Th39</a></span>;<br/>
<a NAME="E8:104"/>
<font color="Maroon" title="c8">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">yy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:104"/>
<font color="Maroon" title="c3">x</font> <a href="gaussint.html#R2" title="GAUSSINT:pred.2">is_associated_to</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:104"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E7:104"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="gaussint.html#T37" target="_self" onmouseover="rs('gaussint/T37')" onmouseout="rh()">Th37</a>, <a class="ref" href="gcd_1.html#T18" onmouseover="rs('gcd_1/T18')" onmouseout="rh()">GCD_1:18</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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