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<div class="add">

<b>let </b><font color="Maroon">I</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">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">EqR1</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">EqR2</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">EqR3</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>









<a NAME="E1:16"/>
for <font color="Olive">EqR4</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>  st <font color="Olive">EqR4</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> holds <br/><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Olive">EqR4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">EqR4</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>

<a NAME="E1:16_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<a NAME="E2:16_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<a NAME="E3:16_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">EqR1</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR2</font> &amp; <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<b>assume </b><a NAME="E5:16_1"/>
<font color="Maroon">EqR4</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font>
 ;<br/>

<a NAME="E6:16_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:16_1"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E8:16_1"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E9:16_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">EqR1</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font> )
 
<b>by </b><i>, , , <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E10:16_1"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18">PBOOLE:18</a></i>;<br/>
<a NAME="E11:16_1"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:16_1"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T18">PBOOLE:18</a></i>;<br/>
<b>hence </b><a NAME="E13:16_1"/>
<font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font>
 
;<br/>
<a NAME="E3:16"/>
for <font color="Olive">EqR4</font> being  <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>  st <font color="Olive">EqR4</font> <a href="pboole.html#R6">=</a> <font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Olive">EqR4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">EqR4</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>

<a NAME="E1:16_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<a NAME="E2:16_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">EqR1</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<a NAME="E3:16_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">EqR2</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR2</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font> &amp; <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR2</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#T16">PBOOLE:16</a></i>;<br/>
<b>assume </b><a NAME="E5:16_2"/>
<font color="Maroon">EqR4</font> <a href="pboole.html#R6">=</a> <font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 ;<br/>

<a NAME="E6:16_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:16_2"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E8:16_2"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E9:16_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">EqR1</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font> )
 
<b>by </b><i>, , , <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E10:16_2"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T18">PBOOLE:18</a></i>;<br/>
<a NAME="E11:16_2"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:16_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="pboole.html#K2">\/</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#T18">PBOOLE:18</a></i>;<br/>
<b>hence </b><a NAME="E13:16_2"/>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR4</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:16"/>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font> <a href="pboole.html#R6">=</a> <font color="Maroon">EqR1</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="msualg_5.html#K4" target="_self">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D13">PBOOLE:def 13</a></i>;<br/>


</div>
