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<b>let </b><font color="Maroon">X</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">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>











<b>assume </b><a NAME="E1:15"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K2" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K2" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16">less_eq_dom</a> <span class="p2">(<span class="default"><a href="mesfunc5.html#K1">R_EAL</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
<font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc6.html#K1" target="_self">less_dom</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="mesfunc1.html#T25">MESFUNC1:25</a></i>;<br/>


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