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<span class="kw">consider </span><font color="Maroon" title="c1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:101_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" onmouseover="rs('xboole_0/S1')" onmouseout="rh()">XBOOLE_0:sch 1</a>();<br/></span>
<a NAME="E3:101_1"/>
<font color="Maroon" title="c1">X</font> is  <a href="membered.html#V1" title="MEMBERED:attr.1">complex-membered</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="membered.html#D1" target="_self" onmouseover="rs('membered/D1')" onmouseout="rh()">MEMBERED:def 1</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">x</font> is  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</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="c2">x</font> is  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a> </span><br/>

<a NAME="E2:101_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:101_1_1"/>
<font color="Maroon" title="c2">x</font> is  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">X</font> = <font color="Maroon" title="c1">X</font> as   <a href="membered.html#V1" title="MEMBERED:attr.1">complex-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">X</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">  for <font color="Olive" title="b1">c</font> being   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</a> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">c</font> be   <a href="xcmplx_0.html#NM1" title="XCMPLX_0:NM.1">Complex</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">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">c</font>] )</span><br/>

<a NAME="E5:101_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" onmouseover="rs('xcmplx_0/D2')" onmouseout="rh()">XCMPLX_0:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:101_1"/>
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">c</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1"><span class="lab"><font color="Green" title="E19">A1</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/>


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