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<span class="kw">let </span><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" 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="c1">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>  implies ( <font color="Maroon" title="c1">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a>  <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>  &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<font color="Maroon" title="c1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<font color="Maroon" title="c1">f</font>) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:54"/>
<font color="Maroon" title="c1">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</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="c1">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a>  <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>  &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<font color="Maroon" title="c1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<font color="Maroon" title="c1">f</font>) )</span><br/>

<a NAME="E2:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A1</font></span>: 
 <a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c1">f</font> <a href="mesfunc5.html#R1" title="MESFUNC5:pred.1">is_integrable_on</a>  <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D4" onmouseover="rs('mesfunc6/D4')" onmouseout="rh()">MESFUNC6:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54"/>
<font color="Maroon" title="c1">f</font> <a href="mesfunc6.html#R2" title="MESFUNC6:pred.2">is_integrable_on</a>  <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun14.html#T38" target="_self" onmouseover="rs('mesfun14/T38')" onmouseout="rh()">Th38</a>, <a class="ref" href="measur12.html#D11" onmouseover="rs('measur12/D11')" onmouseout="rh()">MEASUR12:def 11</a>, <a class="ref" href="measur12.html#D12" onmouseover="rs('measur12/D12')" onmouseout="rh()">MEASUR12:def 12</a>, <a class="ref" href="mesfunc6.html#D4" onmouseover="rs('mesfunc6/D4')" onmouseout="rh()">MESFUNC6:def 4</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">  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<font color="Maroon" title="c1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<font color="Maroon" title="c1">f</font>)</span><br/>

<a NAME="E4:54"/>
(  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<font color="Maroon" title="c1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c1">f</font></span>)</span>) &amp;  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<font color="Maroon" title="c1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc5.html#K7" title="MESFUNC5:func.7">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Maroon" title="c1">f</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc6.html#D3" onmouseover="rs('mesfunc6/D3')" onmouseout="rh()">MESFUNC6:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:54"/>
 <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>,<font color="Maroon" title="c1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc6.html#K1" title="MESFUNC6:func.1">Integral</a> (<a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>,<font color="Maroon" title="c1">f</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="mesfun14.html#T38" target="_self" onmouseover="rs('mesfun14/T38')" onmouseout="rh()">Th38</a>, <a class="ref" href="measur12.html#D11" onmouseover="rs('measur12/D11')" onmouseout="rh()">MEASUR12:def 11</a>, <a class="ref" href="measur12.html#D12" onmouseover="rs('measur12/D12')" onmouseout="rh()">MEASUR12:def 12</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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