<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c1">x</font> is   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> iff <font color="Maroon" title="c1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> )</span><br/>

<a NAME="E1:44"/><span class="lab"><font color="Green" title="E16">L1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> holds  <font color="Olive" title="b1">x</font> is   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="44_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="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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> is   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a></span><br/>

<a NAME="E1:44_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c2">x</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T2" onmouseover="rs('seq_1/T2')" onmouseout="rh()">SEQ_1:2</a>, <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="E2:44_1"/>
<font color="Maroon" title="c2">x</font> is   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_1.html#T2" onmouseover="rs('comseq_1/T2')" onmouseout="rh()">COMSEQ_1:2</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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:44"/>
 for <font color="Olive" title="b1">x</font> being   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> holds  <font color="Olive" title="b1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="44_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</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> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a></span><br/>

<a NAME="E1:44_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c2">x</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</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="comseq_1.html#T1" onmouseover="rs('comseq_1/T1')" onmouseout="rh()">COMSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:44_2"/>
<font color="Maroon" title="c2">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T2" onmouseover="rs('seq_1/T2')" onmouseout="rh()">SEQ_1:2</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:44"/>
( <font color="Maroon" title="c1">x</font> is   <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a> iff <font color="Maroon" title="c1">x</font> is   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E16">L1</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/>


</div>
