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<span class="kw">let </span><font color="Maroon" title="c1">k</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">  for <font color="Olive" title="b1">f</font> being   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">k</font>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">f</font>,<font color="Maroon" title="c1">k</font>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">h</font> being   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c3">h</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<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> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> &amp;  ...  &amp; <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="flexary1.html#T9" target="_self" onmouseover="rs('flexary1/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E3:13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" onmouseover="rs('xreal_1/T232')" onmouseout="rh()">XREAL_1:232</a></span>;<br/>
<a NAME="E4:13"/><span class="kw">then </span>
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/>
<a NAME="E5:13"/><span class="kw">then </span>
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_2.html#T30" onmouseover="rs('rvsum_2/T30')" onmouseout="rh()">RVSUM_2:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:13"/>
(<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>) <a href="flexary1.html#K1" title="FLEXARY1:func.1">+...+</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E9">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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