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<span class="kw">let </span><font color="Maroon" title="c1">x9</font>, <font color="Maroon" title="c2">y9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">y9</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">x9</font>,<font color="Maroon" title="c2">y9</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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">y9</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">x9</font>,<font color="Maroon" title="c2">y9</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/>



<span class="kw">assume </span><a NAME="E1:28"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
( <font color="Maroon" title="c1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">y9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c1">x9</font>,<font color="Maroon" title="c2">y9</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/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:28"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D5" title="XCMPLX_0:def.5">XCMPLX_0:def 5</a></span>;<br/>
<a NAME="E6:28"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="complex1.html#E31"><span class="lab"><font color="Green" title="E16">Lm7</font></span></a></span>;<br/>
<a NAME="E7:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A5</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></span>;<br/>
<a NAME="E8:28"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="complex1.html#E31"><span class="lab"><font color="Green" title="E16">Lm7</font></span></a></span>;<br/>
<a NAME="E9:28"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A7</font></span>: 
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>;<br/>
<a NAME="E11:28"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="complex1.html#E31"><span class="lab"><font color="Green" title="E16">Lm7</font></span></a></span>;<br/>
<span class="kw">hence </span> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> <font color="Maroon" title="c1">x9</font>,<font color="Maroon" title="c2">y9</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E25">A6</font></span></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></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T17" title="ARYTM_0:th.17">ARYTM_0:17</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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