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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</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">  <a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span><br/>

<a NAME="E1:62"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_1"><span class="kw">proof </span></a><div class="add">

 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>

<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<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="E2:62_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:62"/>
 for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="62_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E1:62_2"/>
( <span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">x</font> <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> <font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">x</font> <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> <font color="Maroon" title="c2">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_4.html#T19" target="_self" onmouseover="rs('rvsum_4/T19')" onmouseout="rh()">SCX</a>, <a class="ref" href="rvsum_4.html#T21" target="_self" onmouseover="rs('rvsum_4/T21')" onmouseout="rh()">SFX</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:62_2"/>
<span class="p1">(<span class="default"><a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p2">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:62"/>
 <a href="rvsum_4.html#K7" title="RVSUM_4:func.7">Sequel</a> <font color="Maroon" title="c1">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">f</font> <a href="rvsum_4.html#K3" title="RVSUM_4:func.3">^</a> <span class="p1">(<span class="default"><a href="rvsum_4.html#K1" title="RVSUM_4:func.1">seq_const</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62"><span class="lab"><font color="Green" title="E18">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/>


</div>
