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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="pboole.html#M1">ManySortedSet</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="mssubfam.html#V5">properly-upper-bound</a> <a href="mssubfam.html#NM1">MSSubsetFamily</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="closure1.html#NM1">MSSetOp</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E1:75"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>3</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Olive">b<sub>2</sub></font> iff ( <font color="Olive">b<sub>3</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>3</sub></font> ) ) ) holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:75"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="closure1.html#V1" target="_self">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D2" target="_self">CLOSURE1:def 2</a><br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:75_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="closure1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E3:75_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:75"><i><font color="Green">E30</font></i></a>, <a class="txt" href="closure1.html#E2:75_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E4:75_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E2:75_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:75_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E3:75_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mssubfam.html#T45">MSSUBFAM:45</a></i>;<br/>


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

<b>thus </b><a NAME="E3:75"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="closure1.html#V2" target="_self">monotonic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="pboole.html#M2">Element</a> of  <a href="mssubfam.html#K7">bool</a> <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="closure1.html#D3" target="_self">CLOSURE1:def 3</a><br/>

<b>assume </b><a NAME="E1:75_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:75_2"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>7</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="closure1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:75_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:75"><i><font color="Green">E30</font></i></a>, <a class="txt" href="closure1.html#E3:75_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="relat_1.html#V2">V5</a> <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:75_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>8</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="pboole.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R2">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="closure1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E7:75_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:75_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:75_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:75_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E3:75_2_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="closure1.html#E1:75_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:75_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>10</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>9</sub></font>
 ;<br/>
<a NAME="E6:75_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E3:75_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="closure1.html#E1:75_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="closure1.html#E5:75_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="closure1.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E7:75_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="closure1.html#E6:75_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="closure1.html#E1:75_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="closure1.html#E5:75_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="closure1.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E8:75_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E2:75_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="closure1.html#E6:75_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="closure1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:75_2"/><b>then </b>
 <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K6">meet</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T46">MSSUBFAM:46</a></i>;<br/>
<b>hence </b><a NAME="E9:75_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="closure1.html#K3" target="_self">..</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="closure1.html#E1:75"><i><font color="Green">E30</font></i></a>, <a class="txt" href="closure1.html#E4:75_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="closure1.html#E6:75_2"><i><font color="Green">E34</font></i></a></i>;<br/>


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


</div>
