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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> ex <font color="Olive" title="b3">u</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 <br/>( <font color="Olive" title="b2">y</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="b3">u</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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $1 <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="b3">u</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b2">y</font> );<br/>
<a NAME="E1:87"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> holds <br/> ex <font color="Olive" title="b2">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> implies  ex <font color="Olive" title="b1">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Olive" title="b1">w</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</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">  ex <font color="Olive" title="b1">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Olive" title="b1">w</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">u</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><span class="kw"> such that </span><br/><a NAME="E3:87_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c3">z</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="c4">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D5" onmouseover="rs('quofield/D5')" onmouseout="rh()">QUOFIELD:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">y1</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="E5:87_1"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">y1</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> <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="quofield.html#D1" onmouseover="rs('quofield/D1')" onmouseout="rh()">QUOFIELD:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">y1</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/>
<span class="kw">take </span>
<font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c8">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="c7">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c8">y</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E7:87_1"/>
( <font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c8">y</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E5:87_1"><span class="lab"><font color="Green" title="E37">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">consider </span><font color="Maroon" title="c3">I</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a>,<a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a><span class="kw"> such that </span><br/><a NAME="E3:87"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E35">A1</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E37">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2"><span class="kw">now </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">  for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Olive" title="b2">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z2</font> holds <br/><font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">z1</font>, <font color="Maroon" title="c5">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c4">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c5">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z2</font> implies <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:87_2"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
( <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c5">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z2</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="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z2</font></span><br/><a NAME="E2:87_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A7</font></span>: 
( <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> &amp; <font color="Maroon" title="c5">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">y1</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>, <font color="Maroon" title="c8">u1</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><span class="kw"> such that </span><br/><a NAME="E4:87_2"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
( <font color="Maroon" title="c7">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="c8">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="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">z1</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="c8">u1</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">y2</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>, <font color="Maroon" title="c11">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><span class="kw"> such that </span><br/><a NAME="E6:87_2"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
( <font color="Maroon" title="c10">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="c11">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="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">z2</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="c11">u2</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c10">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E2:87_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E7:87_2"/>
<font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E4:87_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E6:87_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="gaussint.html#E29"><span class="lab"><font color="Green" title="E17">Lm5</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>
<a NAME="E5:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c3">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_3"><span class="kw">now </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">  for <font color="Olive" title="b1">p1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> holds <br/> ex <font color="Olive" title="b2">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">p1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> implies  ex <font color="Olive" title="b1">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:87_3"/>
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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">  ex <font color="Olive" title="b1">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p1</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a><span class="kw"> such that </span><br/><a NAME="E4:87_3"/><span class="lab"><font color="Green" title="E39">A11</font></span>: 
( <font color="Maroon" title="c7">y</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T12" target="_self" onmouseover="rs('gaussint/T12')" onmouseout="rh()">Th12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c7">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/><a NAME="E6:87_3"/>
<font color="Maroon" title="c9">y1</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> <a href="gaussint.html#K6" title="GAUSSINT:func.6">Gauss_INT_Ring</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_3"><span class="lab"><font color="Green" title="E39">A11</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">u1</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">y1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#D1" onmouseover="rs('quofield/D1')" onmouseout="rh()">QUOFIELD:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">z1</font> =  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c10">u1</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">x2</font>, <font color="Maroon" title="c13">y2</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>, <font color="Maroon" title="c14">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><span class="kw"> such that </span><br/><a NAME="E9:87_3"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
( <font color="Maroon" title="c13">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="c14">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="c12">x2</font>,<font color="Maroon" title="c13">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c10">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="c14">u2</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c10">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/><a NAME="E10:87_3"/>
<font color="Maroon" title="c6">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c13">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="gaussint.html#E29"><span class="lab"><font color="Green" title="E17">Lm5</font></span></a>, <a class="txt" href="#E9:87_3"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E4:87_3"><span class="lab"><font color="Green" title="E39">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:87_3"/>
 ex <font color="Olive" title="b1">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_3"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="txt" href="#E4:87_3"><span class="lab"><font color="Green" title="E39">A11</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>
<a NAME="E7:87"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T10" onmouseover="rs('funct_2/T10')" onmouseout="rh()">FUNCT_2:10</a></span>;<br/>
<a NAME="E8:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A13</font></span>: 
<font color="Maroon" title="c3">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E9:87"/><span class="lab"><font color="Green" title="E40">A14</font></span>: 
 for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> holds <br/> ( <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">z2</font></span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">z2</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">z1</font>, <font color="Maroon" title="c5">z2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</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">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z2</font></span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z2</font></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">y1</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>, <font color="Maroon" title="c8">u1</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><span class="kw"> such that </span><br/><a NAME="E2:87_4"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
( <font color="Maroon" title="c7">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="c8">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="c6">x1</font>,<font color="Maroon" title="c7">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">z1</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="c8">u1</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">y2</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>, <font color="Maroon" title="c11">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><span class="kw"> such that </span><br/><a NAME="E4:87_4"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
( <font color="Maroon" title="c10">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="c11">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="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">z2</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="c11">u2</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c10">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">z12</font> = <font color="Maroon" title="c4">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z2</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">y12</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>, <font color="Maroon" title="c15">u12</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><span class="kw"> such that </span><br/><a NAME="E6:87_4"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
( <font color="Maroon" title="c14">y12</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="c15">u12</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="c13">x12</font>,<font color="Maroon" title="c14">y12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z2</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="c15">u12</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x12</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c14">y12</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<a NAME="E7:87_4"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c10">y2</font></span>)</span> <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="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">y1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="xcmplx_1.html#T116" onmouseover="rs('xcmplx_1/T116')" onmouseout="rh()">XCMPLX_1:116</a></span>;<br/>
<a NAME="E8:87_4"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c11">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="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">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="c8">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="c9">x2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">u1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</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="c11">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="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">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="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a></span>;<br/>
<a NAME="E9:87_4"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
 <a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">y1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_4"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<font color="Maroon" title="c4">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z2</font> = 
 <a href="quofield.html#K8" title="QUOFIELD:func.8">qadd</a> (<span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c8">u1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c11">u2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="quofield.html#D12" onmouseover="rs('quofield/D12')" onmouseout="rh()">QUOFIELD:def 12</a></span>
<br/>.= 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T9" onmouseover="rs('quofield/T9')" onmouseout="rh()">QUOFIELD:9</a></span>
;<br/>
<a NAME="E11:87_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A21</font></span>: 
<font color="Maroon" title="c15">u12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_4"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="quofield.html#T5" onmouseover="rs('quofield/T5')" onmouseout="rh()">QUOFIELD:5</a></span>;<br/>
<a NAME="E12:87_4"/><span class="lab"><font color="Green" title="E48">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">u12</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"><span class="p2">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span> <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="c13">x12</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 
<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="#E6:87_4"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E8:87_4"><span class="lab"><font color="Green" title="E45">A19</font></span></a></span>;<br/>
<a NAME="E13:87_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K4" title="QUOFIELD:func.4">padd</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span> <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="c15">u12</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"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y12</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="#E6:87_4"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E9:87_4"><span class="lab"><font color="Green" title="E46">A20</font></span></a></span>;<br/>
<a NAME="E14:87_4"/><span class="kw">then </span>
<font color="Maroon" title="c13">x12</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <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="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_4"><span class="lab"><font color="Green" title="E48">A22</font></span></a>, <a class="txt" href="#E11:87_4"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="ref" href="quofield.html#D4" onmouseover="rs('quofield/D4')" onmouseout="rh()">QUOFIELD:def 4</a></span>;<br/>
<a NAME="E15:87_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A23</font></span>: 
<font color="Maroon" title="c13">x12</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c14">y12</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="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">y1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E6:87_4"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="xcmplx_1.html#T94" onmouseover="rs('xcmplx_1/T94')" onmouseout="rh()">XCMPLX_1:94</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">z12</font> = <font color="Maroon" title="c4">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z2</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">x12</font>, <font color="Maroon" title="c18">y12</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a>, <font color="Maroon" title="c19">u12</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><span class="kw"> such that </span><br/><a NAME="E17:87_4"/><span class="lab"><font color="Green" title="E50">A24</font></span>: 
( <font color="Maroon" title="c18">y12</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="c19">u12</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="c17">x12</font>,<font color="Maroon" title="c18">y12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z2</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="c19">u12</font> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x12</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c18">y12</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<a NAME="E18:87_4"/><span class="lab"><font color="Green" title="E51">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T76" onmouseover="rs('xcmplx_1/T76')" onmouseout="rh()">XCMPLX_1:76</a></span>;<br/>
<a NAME="E19:87_4"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c11">u2</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">u2</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</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="c8">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="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">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="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a></span>;<br/>
<font color="Maroon" title="c4">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z2</font> = 
 <a href="quofield.html#K9" title="QUOFIELD:func.9">qmult</a> (<span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c8">u1</font></span>)</span>,<span class="p1">(<span class="default"><a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <font color="Maroon" title="c11">u2</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="quofield.html#D13" onmouseover="rs('quofield/D13')" onmouseout="rh()">QUOFIELD:def 13</a></span>
<br/>.= 
 <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K5" title="QUOFIELD:func.5">pmult</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quofield.html#T10" onmouseover="rs('quofield/T10')" onmouseout="rh()">QUOFIELD:10</a></span>
;<br/>
<a NAME="E21:87_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A27</font></span>: 
<font color="Maroon" title="c19">u12</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="quofield.html#K10" title="QUOFIELD:func.10">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K5" title="QUOFIELD:func.5">pmult</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:87_4"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="ref" href="quofield.html#T5" onmouseover="rs('quofield/T5')" onmouseout="rh()">QUOFIELD:5</a></span>;<br/>
<a NAME="E22:87_4"/><span class="lab"><font color="Green" title="E54">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">u12</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"><span class="p2">(<span class="default"><a href="quofield.html#K5" title="QUOFIELD:func.5">pmult</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span> <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="c17">x12</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 
<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="#E17:87_4"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E19:87_4"><span class="lab"><font color="Green" title="E52">A26</font></span></a></span>;<br/>
<a NAME="E23:87_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quofield.html#K5" title="QUOFIELD:func.5">pmult</a> (<font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c11">u2</font>)</span>)</span> <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="c19">u12</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">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">y12</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="#E17:87_4"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E19:87_4"><span class="lab"><font color="Green" title="E52">A26</font></span></a></span>;<br/>
<a NAME="E24:87_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">x12</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">y12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:87_4"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="txt" href="#E21:87_4"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="ref" href="quofield.html#D4" onmouseover="rs('quofield/D4')" onmouseout="rh()">QUOFIELD:def 4</a></span>;<br/>
<a NAME="E25:87_4"/><span class="kw">then </span>
<font color="Maroon" title="c17">x12</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c18">y12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E17:87_4"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="ref" href="xcmplx_1.html#T94" onmouseover="rs('xcmplx_1/T94')" onmouseout="rh()">XCMPLX_1:94</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:87_4"/>
( <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z2</font></span>)</span> &amp; <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">z1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="txt" href="#E18:87_4"><span class="lab"><font color="Green" title="E51">A25</font></span></a>, <a class="txt" href="#E7:87_4"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="txt" href="#E17:87_4"><span class="lab"><font color="Green" title="E50">A24</font></span></a>, <a class="txt" href="#E6:87_4"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E2:87_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E4:87_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a>, <a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</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="lab"><font color="Green" title="E41">A29</font></span>: <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span> = 
<font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E40">A14</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E42">A30</font></span>:  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="rlvect_1.html#T28" onmouseover="rs('rlvect_1/T28')" onmouseout="rh()">RLVECT_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T15" onmouseover="rs('rlvect_1/T15')" onmouseout="rh()">RLVECT_1:15</a></span>
<br/>.= 
<font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
;<br/>
<a NAME="E12:87"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A31</font></span>: 
<font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span> <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> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E11:87"><span class="lab"><font color="Green" title="E42">A30</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E44">A32</font></span>: <font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span> = 
<font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E40">A14</font></span></a></span>
;<br/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</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="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E43">A31</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E43">A31</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E16:87"/>
 ex <font color="Olive" title="b1">I</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a>,<a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> st <br/>( (  for <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> holds <br/> ex <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> ex <font color="Olive" title="b5">u</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 <br/>( <font color="Olive" title="b4">y</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="b5">u</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="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">z</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="b5">u</font> &amp; <font color="Olive" title="b1">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Olive" title="b4">y</font> ) ) &amp; <font color="Olive" title="b1">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a> holds <br/> ( <font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font></span>)</span> &amp; <font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font></span>)</span> ) ) &amp; <font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> &amp; <font color="Olive" title="b1">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K7" title="GAUSSINT:func.7">Gauss_INT_Field</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E11:87"><span class="lab"><font color="Green" title="E42">A30</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>
