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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</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="closure1.html#V1" target="_self">reflexive</a> <a href="closure1.html#V2" target="_self">monotonic</a> <a href="closure1.html#NM1">MSSetOp</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="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:69"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a>  <a href="closure1.html#K6" target="_self">MSFixPoints</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="closure1.html#G1" target="_self">MSClosureStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) as    <a href="closure1.html#L1" target="_self">MSClosureStr</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:69"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="closure1.html#V9" target="_self">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="mssubfam.html#V4">absolutely-multiplicative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="mssubfam.html#D6">MSSUBFAM:def 6</a><br/>

<b>assume </b><a NAME="E1:69_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:69_1_1"/>
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E3:69_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R1">in</a>  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mboolean.html#T19">MBOOLEAN:19</a></i>;<br/>
<a NAME="E4:69_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:69_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13">PBOOLE:def 13</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E1:69_1_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:69_1_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="closure1.html#E1:69_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/>
<a NAME="E5:69_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:69_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:69_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:69_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mboolean.html#D1">MBOOLEAN:def 1</a></i>;<br/>
<a NAME="E7:69_1_1_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:69_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:69_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:69_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="closure1.html#E1:69_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="closure1.html#E4:69_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:69_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="closure1.html#E1:69"><i><font color="Green">E25</font></i></a>, <a class="txt" href="closure1.html#E1:69_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="closure1.html#E1:69_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="closure1.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E5:69_1_1_1_1_1"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:69_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mssubfam.html#T2">MSSUBFAM:2</a></i>;<br/><b>hence </b><a NAME="E6:69_1_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:69_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="closure1.html#E1:69_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="closure1.html#E4:69_1_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="closure1.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:69_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E7:69_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="mssubfam.html#T4">MSSUBFAM:4</a></i>;<br/>
<b>hence </b><a NAME="E10:69_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E4:69_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="closure1.html#E5:69_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E2:69_1_1_1"/>
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E3:69_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mssubfam.html#T11">MSSUBFAM:11</a></i>;<br/>
<b>hence </b><a NAME="E3:69_1_1_1"/>
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="closure1.html#K1" target="_self">..</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="closure1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:69_1_1"/>
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:69"><i><font color="Green">E25</font></i></a>, <a class="txt" href="closure1.html#E3:69_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="closure1.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:69_1"/>
the <a href="closure1.html#U1" target="_self">Family</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="mssubfam.html#V4">absolutely-multiplicative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D9" target="_self">CLOSURE1:def 9</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69"/>
 <a href="closure1.html#G1" target="_self">MSClosureStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> #) is   <a href="closure1.html#NM2">MSClosureSystem</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
