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<div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:45_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="prob_1.html#V1">compl-closed</a>
 <b>and </b><br/><a NAME="E2:45_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="prob_1.html#V4">sigma-multiplicative</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="card_4.html#V1">countable</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="measure1.html#D9" target="_self">MEASURE1:def 9</a><br/>

<b>assume </b><a NAME="E3:45_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:45_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D14">SUPINF_2:def 14</a></i>;<br/>
<a NAME="E6:45_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="prob_1.html#K4">Intersection</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="measure1.html#E5:45_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E7:45_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_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="E1:45_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="measure1.html#E5:45_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E2:45_1_1_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="prob_1.html#D4">PROB_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:45_1_1_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure1.html#E1:45_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="measure1.html#E1:45_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="measure1.html#E3:45_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:45_1_1"/><b>then </b>
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="measure1.html#E2:45_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="prob_1.html#D8">PROB_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:45_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure1.html#E6:45_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="measure1.html#E1:45_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="prob_1.html#D1">PROB_1:def 1</a></i>;<br/>


</div>
