<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">y2</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">  for <font color="Olive" title="b1">u1</font>, <font color="Olive" title="b2">u2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a>  st <font color="Maroon" title="c2">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b1">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( <font color="Maroon" title="c1">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">y2</font> iff  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Olive" title="b1">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Olive" title="b2">u2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">u1</font>, <font color="Maroon" title="c6">u2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="quofield.html#K1" title="QUOFIELD:func.1">Q.</a> <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"> ( <font color="Maroon" title="c2">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies ( <font color="Maroon" title="c1">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">y2</font> iff  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
( <font color="Maroon" title="c2">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c4">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"> ( <font color="Maroon" title="c1">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">y2</font> iff  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> )</span><br/>

<a NAME="E2:51"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">u1</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u2</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y2</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>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<a NAME="E3:51"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u2</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y1</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>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<a NAME="E4:51"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">u1</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u2</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u2</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> iff <font color="Maroon" title="c5">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D4" onmouseover="rs('quofield/D4')" onmouseout="rh()">QUOFIELD:def 4</a></span>;<br/>
<a NAME="E5:51"/>
(  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> iff <font color="Maroon" title="c5">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c5">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T5" onmouseover="rs('quofield/T5')" onmouseout="rh()">QUOFIELD:5</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:51_1"/>
(  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> implies <font color="Maroon" title="c5">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T5" onmouseover="rs('quofield/T5')" onmouseout="rh()">QUOFIELD:5</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"> ( <font color="Maroon" title="c5">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> implies  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:51_1"/>
<font color="Maroon" title="c5">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</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">  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font></span><br/>

<a NAME="E4:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:51_1"/>
 <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T8" onmouseover="rs('quofield/T8')" onmouseout="rh()">QUOFIELD:8</a>, <a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</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">hence </span><a NAME="E6:51"/>
( <font color="Maroon" title="c1">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">y2</font> iff  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c5">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c6">u2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T94" onmouseover="rs('xcmplx_1/T94')" onmouseout="rh()">XCMPLX_1:94</a>, <a class="ref" href="xcmplx_1.html#T95" onmouseover="rs('xcmplx_1/T95')" onmouseout="rh()">XCMPLX_1:95</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>
