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<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">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" 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="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c2">g</font>, <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a>  implies <font color="Maroon" title="c1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c2">g</font>, <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:36"/>
<font color="Maroon" title="c1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c2">g</font>, <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="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c2">g</font>, <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">E</font> being    <a href="measur12.html#M3" title="MEASUR12:mode.3">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> <span class="kw"> such that </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
( <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</a></span>;<br/>
<a NAME="E4:36"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">E0</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="measur12.html#M3" title="MEASUR12:mode.3">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="measur12.html#T72" onmouseover="rs('measur12/T72')" onmouseout="rh()">MEASUR12:72</a></span>;<br/>
<a NAME="E6:36"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E</font> <a href="prob_1.html#K6" title="PROB_1:func.6">\/</a> <font color="Maroon" title="c4">E0</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">E1</font> = <font color="Maroon" title="c3">E</font> as    <a href="measur12.html#M3" title="MEASUR12:mode.3">Element</a> of  <a href="measur12.html#K14" title="MEASUR12:func.14">L-Field</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="measure3.html#D3" onmouseover="rs('measure3/D3')" onmouseout="rh()">MEASURE3:def 3</a>, <a class="ref" href="measur12.html#T73" onmouseover="rs('measur12/T73')" onmouseout="rh()">MEASUR12:73</a>, <a class="ref" href="measur12.html#D11" onmouseover="rs('measur12/D11')" onmouseout="rh()">MEASUR12:def 11</a></span>;<br/>
<a NAME="E8:36"/>
<span class="p1">(<span class="default"><a href="measure3.html#K4" title="MEASURE3:func.4">COM</a> <a href="measur12.html#K13" title="MEASUR12:func.13">B-Meas</a></span>)</span> <a href="supinf_2.html#K9" title="SUPINF_2:func.9">.</a> <font color="Maroon" title="c5">E1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E6:36"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="measure3.html#D5" onmouseover="rs('measure3/D5')" onmouseout="rh()">MEASURE3:def 5</a>, <a class="ref" href="measur12.html#T73" onmouseover="rs('measur12/T73')" onmouseout="rh()">MEASUR12:73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:36"/>
<font color="Maroon" title="c1">f</font> <a href="lpspace1.html#R1" title="LPSPACE1:pred.1">a.e.=</a> <font color="Maroon" title="c2">g</font>, <a href="measur12.html#K15" title="MEASUR12:func.15">L-Meas</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="lpspace1.html#D10" onmouseover="rs('lpspace1/D10')" onmouseout="rh()">LPSPACE1:def 10</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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