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<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font>, <font color="Maroon" title="c3">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <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"> for <font color="Olive" title="b1">A</font> being <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">A</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">A</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> holds <br/><span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">g</font>,<font color="Olive" title="b1">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">g</font>,<font color="Olive" title="b1">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g</font>,<font color="Olive" title="b1">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span></span><br/><b>let </b><font color="Maroon" title="c4">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> implies <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> )</span><br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E19">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> or <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> )</span><br/>

<b>assume </b><a NAME="E2:32"/><i><font color="Green" title="E20">A2</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">A</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span></span><br/>

<span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> = 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra5.html#T4" title="INTEGRA5:th.4">INTEGRA5:4</a></i>
<br/>.= 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span>
<b>by </b><i><a class="ref" href="integra9.html#T27" target="_self" title="INTEGRA9:th.27">Th27</a></i>
<br/>.= 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra5.html#T5" title="INTEGRA5:th.5">INTEGRA5:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">integral</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:32"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E2:32"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="integra1.html#T59" title="INTEGRA1:th.59">INTEGRA1:59</a></i>
;<br/>
<b>hence </b><a NAME="E4:32"/>
<span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span>
 ; <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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