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<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> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Olive" title="b1">M</font> holds  <font color="Olive" title="b2">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Olive" title="b3">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <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><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> being  <a href="msualg_5.html#NM1" title="MSUALG_5:NM.1">Equivalence_Relation</a> of <font color="Maroon" title="c2">M</font> holds  <font color="Olive" title="b1">EqR1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Olive" title="b2">EqR2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <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><br/><span class="kw">let </span><font color="Maroon" title="c3">EqR1</font>, <font color="Maroon" title="c4">EqR2</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="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><br/>







<span class="kw">consider </span><font color="Maroon" title="c5">EqR3</font> being   <a href="msualg_4.html#NM3" title="MSUALG_4:NM.3">ManySortedRelation</a> of <font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<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="pboole.html#K2" title="PBOOLE:func.2">\/</a> <font color="Maroon" title="c4">EqR2</font>
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E9">A2</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#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_5.html#K3" title="MSUALG_5:func.3">EqCl</a> <font color="Maroon" title="c5">EqR3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D4" target="_self" title="MSUALG_5:def.4">Def4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</font> be    <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"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c5">EqR3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_1"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"> <font color="Maroon" title="c5">EqR3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i9</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><a NAME="E3:12_1"/>
<font color="Maroon" title="c5">EqR3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">EqR3</font> <a href="msualg_4.html#K1" title="MSUALG_4:func.1">.</a> <font color="Maroon" title="c7">i9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D1" target="_self" title="MSUALG_5:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:12_1"/>
<font color="Maroon" title="c5">EqR3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="msualg_5.html#D3" target="_self" title="MSUALG_5:def.3">Def3</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>
<span class="kw">hence </span><a NAME="E5:12"/>
<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="txt" href="#E2:12"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</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/>


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