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<b>let </b><font color="Maroon" title="c1">f</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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</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; ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="integra9.html#K2" title="INTEGRA9:func.2">||..</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">A</font></span><a href="integra9.html#K2" title="INTEGRA9:func.2">..||</a></span></span><br/><b>let </b><font color="Maroon" title="c2">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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="integra9.html#K2" title="INTEGRA9:func.2">||..</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span><a href="integra9.html#K2" title="INTEGRA9:func.2">..||</a></span> )</span><br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E22">A1</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> or  ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="integra9.html#K2" title="INTEGRA9:func.2">||..</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span><a href="integra9.html#K2" title="INTEGRA9:func.2">..||</a></span> )</span><br/>

<b>assume </b><a NAME="E2:50"/><i><font color="Green" title="E23">A2</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">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">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font> is <a href="integra1.html#V4" title="INTEGRA1:attr.4">integrable</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="integra5.html#K1" title="INTEGRA5:func.1">||</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="integra9.html#K2" title="INTEGRA9:func.2">||..</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span><a href="integra9.html#K2" title="INTEGRA9:func.2">..||</a></span></span><br/>

<a NAME="E3:50"/>
<span class="p1"><a href="integra9.html#K1" title="INTEGRA9:func.1">|||(</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span><a href="integra9.html#K1" title="INTEGRA9:func.1">)|||</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E2:50"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="integra2.html#T32" title="INTEGRA2:th.32">INTEGRA2:32</a></i>;<br/>
<b>hence </b><a NAME="E4:50"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="integra9.html#K2" title="INTEGRA9:func.2">||..</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">A</font></span><a href="integra9.html#K2" title="INTEGRA9:func.2">..||</a></span>
 <b>by </b><i><a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></i>; <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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