<?xml version="1.0"?>
<div class="add">

<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">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="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K3">RAT</a> ,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc2.html#R1" target="_self">is_finite</a> 
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">g</font> <a href="mesfunc2.html#R1" target="_self">is_finite</a> 
 <b>and </b><br/><a NAME="E3:4"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">p</font> being  <a href="rat_1.html#NM1">Rational</a> holds  <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</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">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</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">g</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:4"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:4_1"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:4_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E5:4_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/>
<a NAME="E6:4_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:4_1"/><b>then </b>
( <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:4_1"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/>
<a NAME="E9:4_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="extreal2.html#T27">EXTREAL2:27</a></i>;<br/>
<a NAME="E10:4_1"/><i><font color="Green">E46</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="extreal1.html#T4">EXTREAL1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E13:4_1"/>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/>
<a NAME="E14:4_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g1</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="rat_1.html#NM1">Rational</a><b> such that </b><br/><a NAME="E16:4_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/>
<a NAME="E17:4_1"/>
( not <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f1</font> &amp; not <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g1</font> )
 
<b>by </b><i>, <a class="ref" href="real_2.html#T165">REAL_2:165</a></i>;<br/>
<a NAME="E18:4_1"/><b>then </b>
( not  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; not  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> )
 
;<br/>
<a NAME="E19:4_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<a NAME="E20:4_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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">g</font>,<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="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E21:4_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</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">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</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">g</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:4_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="rat_1.html#D2">RAT_1:def 2</a></i>;<br/>
<a NAME="E23:4_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E24:4_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:4_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 ;<br/>
<b>thus </b><a NAME="E27:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:4"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:4_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="rat_1.html#NM1">Rational</a> <b>by </b><i>, <a class="ref" href="rat_1.html#D2">RAT_1:def 2</a></i>;<br/>
<a NAME="E7:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</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">f</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</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">g</font>,<span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:4_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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">f</font>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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">g</font>,<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="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:4_2"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <font color="Maroon">g</font>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:4_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> ) &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp;  ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E13:4_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">y2</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E15:4_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<a NAME="E16:4_2"/>
( <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y1</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y2</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i>, , , , , </i>;<br/>
<a NAME="E17:4_2"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="extreal2.html#T58">EXTREAL2:58</a></i>;<br/>
<a NAME="E18:4_2"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="extreal1.html#T4">EXTREAL1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">y1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">y2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/>
<a NAME="E21:4_2"/>
( <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:4_2"/><b>then </b>
( <font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g1</font> )
 
<b>by </b><i><a class="ref" href="real_2.html#T165">REAL_2:165</a></i>;<br/>
<a NAME="E23:4_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:4_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">f1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E25:4_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>
<br/>.= 
<font color="Maroon">f1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">g1</font>
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
;<br/>
<a NAME="E27:4_2"/><b>then </b>
not  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<a NAME="E28:4_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K15">less_dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mesfunc1.html#D12">MESFUNC1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E29:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:4"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K15">less_dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K13">+</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
