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<a NAME="E1:78_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="78_1_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> ; <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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="E1:78_1_1_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E3:78_1_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E4:78_1_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_4.html#D9" target="_self" onmouseover="rs('rvsum_4/D9')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E5:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">Segm</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a></span>;<br/>
<a NAME="E6:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton02.html#T106" onmouseover="rs('newton02/T106')" onmouseout="rh()">NEWTON02:106</a></span>;<br/>
<a NAME="E7:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:78_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:78_1_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="rvsum_4.html#D9" target="_self" onmouseover="rs('rvsum_4/D9')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:78_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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">hence </span><a NAME="E2:78_1_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a></span>)</span> <a href="rvsum_4.html#K8" title="RVSUM_4:func.8">^</a> <font color="Maroon" title="c1">f</font> is  <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_0.html#D7" onmouseover="rs('valued_0/D7')" onmouseout="rh()">VALUED_0:def 7</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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