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<b>thus </b><a NAME="E1:43_2_1"/>
( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  implies  ex <font color="Olive" title="b1">IT</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">z'</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z'</font> &amp; <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Olive" title="b2">z'</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2_1_1"/>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">s</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="E3:43_2_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c1">z</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="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">z'</font> = <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>;<br/>
<b>take </b>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
;<br/>

<b>take </b>
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
;<br/>

<b>thus </b><a NAME="E4:43_2_1_1"/>
( <font color="Maroon" title="c1">z</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="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> )
 <b>by </b><i><a class="txt" href="quaterni.html#E3:43_2_1_1"><i><font color="Green" title="E21">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:43_2_1"/><i><font color="Green" title="E21">A4</font></i>: 
not <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 ;<br/>

<a NAME="E3:43_2_1"/>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a> 
 
<b>by </b><i><a class="ref" href="quaterni.html#D2" target="_self" title="QUATERNI:def.2">Def2</a></i>;<br/>
<a NAME="E4:43_2_1"/><b>then </b>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="quaterni.html#E2:43_2_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">z</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 4, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
;<br/>

<b>take </b>
<font color="Maroon" title="c2">f</font>
;<br/>

<b>thus </b><a NAME="E6:43_2_1"/>
( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 )
 ;<br/>


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