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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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" 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> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><br/>  for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> holds <br/>( <font color="Olive" title="b2">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Olive" title="b4">B</font> iff <font color="Olive" title="b2">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Olive" title="b3">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b4">B</font> )</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" 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> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Olive" title="b3">B</font> iff <font color="Olive" title="b1">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Olive" title="b2">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b3">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" 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>, <font color="Olive" title="b2">B</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font>  st  <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> <font color="Olive" title="b1">A</font> holds <br/>( <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Olive" title="b2">B</font> iff <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Olive" title="b1">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Olive" title="b2">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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> <font color="Maroon" title="c4">A</font> implies ( <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font> iff <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 <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> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" 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="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font> iff <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font> )</span><br/>

<a NAME="E2:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_3.html#D3" title="COMSEQ_3:def.3">COMSEQ_3:def 3</a></span>;<br/>
<a NAME="E3:17"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="comseq_3.html#D4" title="COMSEQ_3:def.4">COMSEQ_3:def 4</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" 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="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font> implies <font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" 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="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font></span><br/><a NAME="E2:17_1"/><span class="kw">then </span>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D1" target="_self" title="MESFUN6C:def.1">Def3</a></span>;<br/><a NAME="E3:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="mesfunc6.html#T80" title="MESFUNC6:th.80">MESFUNC6:80</a></span>;<br/><a NAME="E4:17_1"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="mesfun6c.html#D1" target="_self" title="MESFUN6C:def.1">Def3</a></span>;<br/><a NAME="E5:17_1"/><span class="kw">then </span>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="mesfunc6.html#T80" title="MESFUNC6:th.80">MESFUNC6:80</a></span>;<br/><span class="kw">hence </span><a NAME="E6:17_1"/>
<font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="mesfun6c.html#D1" target="_self" title="MESFUN6C:def.1">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" 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="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font></span><br/><a NAME="E2:17_2"/><span class="kw">then </span>
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfun6c.html#D1" target="_self" title="MESFUN6C:def.1">Def3</a></span>;<br/><a NAME="E3:17_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="mesfunc6.html#T80" title="MESFUNC6:th.80">MESFUNC6:80</a></span>;<br/><a NAME="E4:17_2"/>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c4">A</font> <a href="prob_1.html#K5" title="PROB_1:func.5">/\</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="mesfun6c.html#D1" target="_self" title="MESFUN6C:def.1">Def3</a></span>;<br/><a NAME="E5:17_2"/><span class="kw">then </span>
 <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f</font> <a href="mesfunc6.html#R1" title="MESFUNC6:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="mesfunc6.html#T80" title="MESFUNC6:th.80">MESFUNC6:80</a></span>;<br/><span class="kw">hence </span><a NAME="E6:17_2"/>
<font color="Maroon" title="c3">f</font> <a href="mesfun6c.html#R1" title="MESFUN6C:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="mesfun6c.html#D1" target="_self" title="MESFUN6C:def.1">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>


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