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<span class="kw">consider </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">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="E2:6_4_1"/><span class="lab"><font color="Green" title="E3">A10</font></span>: 
<font color="Maroon" title="c1">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="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:6_4_1"/><span class="lab"><font color="Green" title="E4">A11</font></span>: 
<font color="Maroon" title="c2">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="c5">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:6_4_1"/><span class="lab"><font color="Green" title="E5">A12</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">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="c3">x1</font>,<font color="Maroon" title="c5">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="c4">x2</font>,<font color="Maroon" title="c6">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="c3">x1</font>,<font color="Maroon" title="c6">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="c4">x2</font>,<font color="Maroon" title="c5">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="E5:6_4_1"/>
<font color="Maroon" title="c4">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="#E2:6_4_1"><span class="lab"><font color="Green" title="E3">A10</font></span></a>, <a class="txt" href="xreal_0.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>;<br/>
<a NAME="E6:6_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A13</font></span>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">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#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E7:6_4_1"/><span class="lab"><font color="Green" title="E7">A14</font></span>: 
<font color="Maroon" title="c6">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="#E3:6_4_1"><span class="lab"><font color="Green" title="E4">A11</font></span></a>, <a class="txt" href="xreal_0.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>;<br/>
<a NAME="E8:6_4_1"/><span class="kw">then </span>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<font color="Maroon" title="c6">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#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E9:6_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A15</font></span>: 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</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="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>;<br/>
<a NAME="E10:6_4_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c6">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="#E7:6_4_1"><span class="lab"><font color="Green" title="E7">A14</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/>
<a NAME="E11:6_4_1"/><span class="kw">then </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="c3">x1</font>,<font color="Maroon" title="c6">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="c4">x2</font>,<font color="Maroon" title="c5">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="#E6:6_4_1"><span class="lab"><font color="Green" title="E6">A13</font></span></a>, <a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <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="c3">x1</font>,<font color="Maroon" title="c5">y1</font>)</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:6_4_1"><span class="lab"><font color="Green" title="E5">A12</font></span></a>, <a class="txt" href="#E9:6_4_1"><span class="lab"><font color="Green" title="E8">A15</font></span></a>, <a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:6_4_1"/>
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">y</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" target="_self" title="XREAL_0:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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