<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">I</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">EqR1</font>, <font color="Olive" title="b3">EqR2</font>, <font color="Olive" title="b4">EqR3</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Olive" title="b1">M</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Olive" title="b3">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Olive" title="b4">EqR3</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Olive" title="b4">EqR3</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">EqR1</font>, <font color="Olive" title="b2">EqR2</font>, <font color="Olive" title="b3">EqR3</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Olive" title="b2">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Olive" title="b3">EqR3</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Olive" title="b3">EqR3</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">EqR1</font>, <font color="Maroon" title="c4">EqR2</font>, <font color="Maroon" title="c5">EqR3</font> be   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span></span><br/>









<a NAME="E1:16"/>
for <font color="Olive" title="b1">EqR4</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">EqR4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">EqR4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">EqR4</font> be   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">EqR4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font> )</span><br/>

<a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T6" target="_self" title="MSUALG_5:th.6">Th6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:16_1"/>
<font color="Maroon" title="c6">EqR4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font></span><br/>

<a NAME="E3:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T6" target="_self" title="MSUALG_5:th.6">Th6</a></span>;<br/>
<a NAME="E4:16_1"/>
<font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E5:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E6:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E7:16_1"/>
<font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E8:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E9:16_1"/>
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E10:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E11:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<a NAME="E12:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T7" target="_self" title="MSUALG_5:th.7">Th7</a></span>;<br/>
<a NAME="E13:16_1"/>
<font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E14:16_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E15:16_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:16_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T7" target="_self" title="MSUALG_5:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span>
 
;<br/>
<a NAME="E3:16"/>
for <font color="Olive" title="b1">EqR4</font> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">EqR4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> holds <br/><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">EqR4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">EqR4</font> be   <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">EqR4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> implies <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font> )</span><br/>

<a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T6" target="_self" title="MSUALG_5:th.6">Th6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:16_2"/>
<font color="Maroon" title="c6">EqR4</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font></span><br/>

<a NAME="E3:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T6" target="_self" title="MSUALG_5:th.6">Th6</a></span>;<br/>
<a NAME="E4:16_2"/>
<font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E5:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E6:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E7:16_2"/>
<font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E8:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E9:16_2"/>
<font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">EqR2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E10:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E11:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">EqR2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<a NAME="E12:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T7" target="_self" title="MSUALG_5:th.7">Th7</a></span>;<br/>
<a NAME="E13:16_2"/>
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">EqR2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T16" title="PBOOLE:th.16">PBOOLE:16</a></span>;<br/>
<a NAME="E14:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></span>;<br/>
<a NAME="E15:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="pboole.html#T18" title="PBOOLE:th.18">PBOOLE:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:16_2"/>
<font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c6">EqR4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#T7" target="_self" title="MSUALG_5:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:16"/><span class="kw">then </span>
<font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c4">EqR2</font></span>)</span> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">EqR1</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">EqR2</font> <a href="msualg_5.html#K4" title="MSUALG_5:func.4">"\/"</a> <font color="Maroon" title="c5">EqR3</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
