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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="measure1.html#M3">sigma_Measure</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>









<b>assume </b><a NAME="E1:68"/><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:68_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> iff ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E2:68_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> iff <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/></div><b>end;</b></div><a NAME="E2:68_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E3:68_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:68_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:68"><i><font color="Green">E66</font></i></a></i>
;<br/><b>hence </b><a NAME="E2:68_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a>, <a class="txt" href="mesfunc5.html#E1:68_2"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:68_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:68"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="measure1.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc5.html#E2:68"><i><font color="Green">E66</font></i></a></i>
;<br/><b>hence </b><a NAME="E2:68_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="mesfunc1.html#R1">is_measurable_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a>, <a class="txt" href="mesfunc5.html#E4:68"><i><font color="Green">E67</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:68"/>
<font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#R2">is_measurable_on</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17">MESFUNC1:def 17</a></i>;<br/>


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