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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds   <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Olive" title="b2">f2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="rfunct_3.html#K4" title="RFUNCT_3:func.4">PFuncs</a> <font color="Maroon" title="c1">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">f2</font></span>)</span></span><br/>



<b>set </b><font color="Maroon" title="c4">o</font> =  <a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font>;<br/>
<a NAME="E1:35"/>
(  <a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp;  <a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp;  <a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font> is <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T16" target="_self" title="RFUNCT_3:th.16">Th16</a>, <a class="ref" href="rfunct_3.html#T17" target="_self" title="RFUNCT_3:th.17">Th17</a>, <a class="ref" href="rfunct_3.html#T20" target="_self" title="RFUNCT_3:th.20">Th20</a></i>;<br/>
<b>hence </b> <a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K14" title="RFUNCT_3:func.14">addpfunc</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c2">f1</font></span>)</span>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T6" title="FINSOP_1:th.6">FINSOP_1:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c2">f1</font></span>)</span> <a href="rfunct_3.html#K6" title="RFUNCT_3:func.6">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K15" title="RFUNCT_3:func.15">Sum</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self" title="RFUNCT_3:def.4">Def4</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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