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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <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>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E2:56"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="measure1.html#T57" target="_self">Th57</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E3:56"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> holds  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:56"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:56"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 <b>by </b><i><a class="ref" href="measure1.html#D4" target="_self">Def4</a></i>;<br/>

<a NAME="E5:56"/>
 <a href="xboole_0.html#K1">{}</a>  is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T10">PROB_1:10</a></i>;<br/>
<b>hence </b><a NAME="E6:56"/>
<font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K13">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="measure1.html#E2:56"><i><font color="Green">E31</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="measure1.html#NM4">Sep_Sequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E7:56"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="measure1.html#K5" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="measure1.html#T53" target="_self">Th53</a></i>;<br/>
<a NAME="E8:56"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K5" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i><a class="txt" href="measure1.html#E2:56"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E9:56"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V6">nonnegative</a>
 
<b>by </b><i><a class="txt" href="measure1.html#E4:56"><i><font color="Green">E32</font></i></a>, <a class="ref" href="measure1.html#T54" target="_self">Th54</a></i>;<br/>
<a NAME="E10:56"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="measure1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:56_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:56_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="measure1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="measure1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E3:56_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="measure1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i><a class="txt" href="measure1.html#E2:56"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:56"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1" target="_self">.</a> 0
 
<b>by </b><i><a class="txt" href="measure1.html#E9:56"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_2.html#T67">SUPINF_2:67</a></i>;<br/>
<a NAME="E12:56"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K18">Ser</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="measure1.html#K1" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="measure1.html#K1" target="_self">.</a> 0
 
<b>by </b><i><a class="ref" href="supinf_2.html#T63">SUPINF_2:63</a></i>;<br/>
<b>hence </b><a NAME="E13:56"/>
 <a href="supinf_2.html#K19">SUM</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K13">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <span class="p2">(<span class="default"><a href="measure1.html#K5" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="measure1.html#E8:56"><i><font color="Green">E33</font></i></a>, <a class="txt" href="measure1.html#E10:56"><i><font color="Green">E35</font></i></a>, <a class="txt" href="measure1.html#E11:56"><i><font color="Green">E36</font></i></a></i>;<br/>


</div>
