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<b>let </b><font color="Maroon" title="c1">x'</font>, <font color="Maroon" title="c2">y'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c1">x'</font>,<font color="Maroon" title="c2">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</font></span><br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> implies  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c1">x'</font>,<font color="Maroon" title="c2">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font> )</span><br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c1">x'</font>,<font color="Maroon" title="c2">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">y1</font>, <font color="Maroon" title="c8">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:20"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c8">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>and </b><br/><a NAME="E5:20"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y1</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D5" title="XCMPLX_0:def.5">XCMPLX_0:def 5</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> )
 
<b>by </b><i><a class="txt" href="complex1.html#E3:20"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="complex1.html#E4:20"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="complex1.html#E32"><i><font color="Green" title="E16">Lm7</font></i></a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c8">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="complex1.html#E3:20"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="complex1.html#E4:20"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="complex1.html#E32"><i><font color="Green" title="E16">Lm7</font></i></a></i>;<br/>
<a NAME="E8:20"/><b>then </b>
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>;<br/>
<a NAME="E9:20"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></i>;<br/>
<b>thus </b> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c1">x'</font>,<font color="Maroon" title="c2">y'</font> = 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c6">x2</font></span>)</span>,<font color="Maroon" title="c8">y2</font></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E1:20"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="complex1.html#E6:20"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="complex1.html#E7:20"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a>, <a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></i>
<br/>.= 
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c6">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17" title="ARYTM_0:th.17">ARYTM_0:17</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">y</font>
<b>by </b><i><a class="txt" href="complex1.html#E5:20"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="complex1.html#E9:20"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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