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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">S</font> being   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">V179</a>() <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b4">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">f</font> is <font color="Olive" title="b4">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b3">g</font> is <font color="Olive" title="b4">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Olive" title="b3">g</font></span>)</span> holds <br/><font color="Olive" title="b2">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Olive" title="b3">g</font> is <font color="Olive" title="b4">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c1">X</font>; <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">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">V179</a>() <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a><br/>  for <font color="Olive" title="b3">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">f</font> is <font color="Olive" title="b3">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b2">g</font> is <font color="Olive" title="b3">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Olive" title="b2">g</font></span>)</span> holds <br/><font color="Olive" title="b1">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Olive" title="b2">g</font> is <font color="Olive" title="b3">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font> be  <a href="mesfunc5.html#V4" title="MESFUNC5:attr.4">V179</a>() <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</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">  for <font color="Olive" title="b1">A</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">f</font> is <font color="Olive" title="b1">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c4">g</font> is <font color="Olive" title="b1">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font></span>)</span> holds <br/><font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font> is <font color="Olive" title="b1">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>; <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="c3">f</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c4">g</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font></span>)</span> implies <font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:89"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
<font color="Maroon" title="c3">f</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">and </span><br/><a NAME="E2:89"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon" title="c4">g</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">and </span><br/><a NAME="E3:89"/><span class="lab"><font color="Green" title="E63">A5</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font></span>)</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"> <font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> </span><br/>

<a NAME="E4:89"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc9.html#T1" onmouseover="rs('mesfunc9/T1')" onmouseout="rh()">MESFUNC9:1</a></span>;<br/>
<a NAME="E5:89"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/>
<a NAME="E6:89"/><span class="kw">then </span>
( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E4:89"><span class="lab"><font color="Green" title="E64">A6</font></span></a></span>;<br/>
<a NAME="E7:89"/><span class="kw">then </span>
(  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">f</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  &amp;  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c4">g</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E2:89"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="measur11.html#T63" target="_self" onmouseover="rs('measur11/T63')" onmouseout="rh()">Th59</a></span>;<br/>
<a NAME="E8:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A7</font></span>: 
<span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c4">g</font></span>)</span> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T31" onmouseover="rs('mesfunc5/T31')" onmouseout="rh()">MESFUNC5:31</a></span>;<br/>
<a NAME="E9:89"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c4">g</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc1.html#D7" onmouseover="rs('mesfunc1/D7')" onmouseout="rh()">MESFUNC1:def 7</a></span>;<br/>
<span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#T16" onmouseover="rs('mesfunc5/T16')" onmouseout="rh()">MESFUNC5:16</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc9.html#T1" onmouseover="rs('mesfunc9/T1')" onmouseout="rh()">MESFUNC9:1</a></span>
;<br/>
<a NAME="E11:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A8</font></span>: 
 <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="txt" href="#E8:89"><span class="lab"><font color="Green" title="E65">A7</font></span></a>, <a class="ref" href="measur11.html#T63" target="_self" onmouseover="rs('measur11/T63')" onmouseout="rh()">Th59</a></span>;<br/>
<a NAME="E12:89"/>
<span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="measur11.html#K14" title="MEASUR11:func.14">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="mesfunc1.html#K7" title="MESFUNC1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="measur11.html#T64" target="_self" onmouseover="rs('measur11/T64')" onmouseout="rh()">Th60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:89"/>
<font color="Maroon" title="c3">f</font> <a href="measur11.html#K13" title="MEASUR11:func.13">+</a> <font color="Maroon" title="c4">g</font> is <font color="Maroon" title="c5">A</font> <a href="mesfunc1.html#V1" title="MESFUNC1:attr.1">-measurable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89"><span class="lab"><font color="Green" title="E66">A8</font></span></a>, <a class="ref" href="dblseq_3.html#T2" onmouseover="rs('dblseq_3/T2')" onmouseout="rh()">DBLSEQ_3:2</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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