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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">BL</font> be   <a href="lattices.html#V17" title="LATTICES:attr.17">Boolean</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">AF</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#NM2" title="LATTICE4:NM.2">ClosedSubset</a> of <font color="Maroon" title="c1">BL</font>  st  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">AF</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">AF</font> holds <br/> for <font color="Olive" title="b2">B</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Olive" title="b1">AF</font> holds <br/> ex <font color="Olive" title="b3">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b3">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Olive" title="b1">AF</font> &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b3">B0</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">AF</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="lattice4.html#NM2" title="LATTICE4:NM.2">ClosedSubset</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> implies  for <font color="Olive" title="b1">B</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> holds <br/> ex <font color="Olive" title="b2">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b2">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b2">B0</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font>
 <span class="kw">and </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> holds <br/> ex <font color="Olive" title="b2">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b2">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b2">B0</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">C</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> ;<br/>
<a NAME="E3:53"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">A1</font>, <font color="Maroon" title="c6">A2</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E3:53_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c5">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c6">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:53_1"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
( <font color="Maroon" title="c5">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c6">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E6:53_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c6">A2</font> &amp; <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c5">A1</font> )
 ;<br/>
<a NAME="E7:53_1"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c8">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E6:53_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="lattice4.html#T41" target="_self" title="LATTICE4:th.41">Th48</a></span>;<br/>
<a NAME="E8:53_1"/>
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="txt" href="#E6:53_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="ref" href="lattice4.html#T17" target="_self" title="LATTICE4:th.17">Th24</a>, <a class="ref" href="lattice4.html#T24" target="_self" title="LATTICE4:th.24">Th31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="lattice4.html#T37" target="_self" title="LATTICE4:th.37">Th44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>] <span class="kw">means </span>( $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> implies  ex <font color="Olive" title="b1">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b3">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> ($1,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> ) );<br/>
<span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> implies  ex <font color="Olive" title="b1">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:53"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> )</span><br/>

<a NAME="E5:53"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
 for <font color="Olive" title="b1">B9</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><br/>  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B9</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">J</font> =  the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">B9</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B9</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B9</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:53_2"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
( <font color="Maroon" title="c6">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> implies  ex <font color="Olive" title="b1">B1</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b3">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c6">B9</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B1</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

<span class="kw">assume </span><a NAME="E2:53_2"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<font color="Maroon" title="c6">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b3">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">B1</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E4:53_2"/><span class="lab"><font color="Green" title="E49">A12</font></span>: 
<font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E5:53_2"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
 <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c6">B9</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_2"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="setwiseo.html#T5" title="SETWISEO:th.5">SETWISEO:5</a></span>;<br/>
<a NAME="E6:53_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_2"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="setwiseo.html#T5" title="SETWISEO:th.5">SETWISEO:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E8:53_2"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c10">q</font>
 <span class="kw">and </span><br/><a NAME="E9:53_2"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font>
 <span class="kw">and </span><br/><a NAME="E10:53_2"/><span class="lab"><font color="Green" title="E53">A16</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">AF</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T37" target="_self" title="LATTICE4:th.37">Th44</a></span>;<br/>
<a NAME="E11:53_2"/><span class="lab"><font color="Green" title="E54">A17</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> holds <br/> ex <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>  st <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Olive" title="b1">b</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> holds <br/> ex <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b1">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c12">b</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> implies  ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">b</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:53_2_1"/><span class="lab"><font color="Green" title="E55">A18</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c12">b</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">b</font> )</span><br/>

<a NAME="E2:53_2_1"/>
<span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c12">b</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T36" title="FUNCT_2:th.36">FUNCT_2:36</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c11">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_2_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E5:53_2_1"/><span class="lab"><font color="Green" title="E56">A19</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font>
 <span class="kw">and </span><br/><a NAME="E6:53_2_1"/><span class="lab"><font color="Green" title="E57">A20</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c12">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_2_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></span>;<br/>
<font color="Maroon" title="c13">x</font> = 
 the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span>,<font color="Maroon" title="c12">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53_2_1"><span class="lab"><font color="Green" title="E57">A20</font></span></a>, <a class="ref" href="funcop_1.html#T48" title="FUNCOP_1:th.48">FUNCOP_1:48</a></span>
<br/>.= 
<font color="Maroon" title="c14">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:53_2_1"/>
 ex <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c12">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_2_1"><span class="lab"><font color="Green" title="E56">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:53_2"/><span class="lab"><font color="Green" title="E55">A21</font></span>: 
<span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E3:53_2_2"/><span class="lab"><font color="Green" title="E56">A22</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font>
 <span class="kw">and </span><br/><a NAME="E4:53_2_2"/><span class="lab"><font color="Green" title="E57">A23</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_2"><span class="lab"><font color="Green" title="E54">A17</font></span></a></span>;<br/>
<a NAME="E5:53_2_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_2"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E3:53_2_2"><span class="lab"><font color="Green" title="E56">A22</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">A1</font>, <font color="Maroon" title="c14">A2</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E7:53_2_2"/><span class="lab"><font color="Green" title="E58">A24</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c14">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:53_2_2"/><span class="lab"><font color="Green" title="E59">A25</font></span>: 
( <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<a NAME="E9:53_2_2"/>
 ex <font color="Olive" title="b1">A19</font>, <font color="Olive" title="b2">A29</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Olive" title="b1">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c15">A19</font> = <font color="Maroon" title="c13">A1</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">p</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A29</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b1">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c15">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b1">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c16">A29</font> = <font color="Maroon" title="c14">A2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c16">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c15">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c16">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c14">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_2_2"><span class="lab"><font color="Green" title="E57">A23</font></span></a>, <a class="txt" href="#E7:53_2_2"><span class="lab"><font color="Green" title="E58">A24</font></span></a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c16">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T14" target="_self" title="LATTICE4:th.14">Th21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:53_2_2_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c16">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c15">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c16">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:53_2"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E8:53_2_2"><span class="lab"><font color="Green" title="E59">A25</font></span></a>, <a class="ref" href="setwiseo.html#T5" title="SETWISEO:th.5">SETWISEO:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:53_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:53_2"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
<span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_2_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E3:53_2_3"/><span class="lab"><font color="Green" title="E57">A27</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B1</font>
 <span class="kw">and </span><br/><a NAME="E4:53_2_3"/><span class="lab"><font color="Green" title="E58">A28</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:53_2"><span class="lab"><font color="Green" title="E54">A17</font></span></a></span>;<br/>
<a NAME="E5:53_2_3"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_2"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E3:53_2_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">A1</font>, <font color="Maroon" title="c14">A2</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font><span class="kw"> such that </span><br/><a NAME="E7:53_2_3"/><span class="lab"><font color="Green" title="E59">A29</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c14">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:53_2_3"/><span class="lab"><font color="Green" title="E60">A30</font></span>: 
( <font color="Maroon" title="c13">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c14">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 ;<br/>
<a NAME="E9:53_2_3"/>
 ex <font color="Olive" title="b1">A19</font>, <font color="Olive" title="b2">A29</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Olive" title="b1">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c15">A19</font> = <font color="Maroon" title="c13">A1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A29</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b1">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c15">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b1">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c16">A29</font> = <font color="Maroon" title="c14">A2</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c10">q</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c16">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c15">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c16">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c13">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c14">A2</font>,<span class="p3">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_2_3"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="txt" href="#E7:53_2_3"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c16">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T38" target="_self" title="LATTICE4:th.38">Th45</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:53_2_3_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c15">A19</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c16">A29</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c15">A19</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c16">A29</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:53_2"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E8:53_2_3"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="setwiseo.html#T5" title="SETWISEO:th.5">SETWISEO:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:53_2_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span> )</span><br/>

<font color="Maroon" title="c7">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:53_2"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="lattices.html#T24" title="LATTICES:th.24">LATTICES:24</a></span>
<br/>.= 
<font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></span>
;<br/>
<span class="kw">then </span> <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_2"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="ref" href="lattice4.html#T38" target="_self" title="LATTICE4:th.38">Th45</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T11" title="LATTICES:th.11">LATTICES:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T25" target="_self" title="LATTICE4:th.25">Th32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T25" target="_self" title="LATTICE4:th.25">Th32</a></span>
<br/>.= 
 <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T23" target="_self" title="LATTICE4:th.23">Th30</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:53_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p3">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">B9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<font color="Maroon" title="c9">p</font>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"> the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">BL</font> <a href="funcop_1.html#K9" title="FUNCOP_1:func.9">[:]</a> (<span class="p4">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>)</span>)</span> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c8">B1</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:53_2"><span class="lab"><font color="Green" title="E55">A21</font></span></a>, <a class="txt" href="#E13:53_2"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:53"/><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:53_3"/>
 <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b3">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c5">B0</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c5">B0</font> )</span><br/>

<a NAME="E2:53_3"/><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<font color="Maroon" title="c5">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> </span><br/>

<a NAME="E2:53_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:53_3_1"/>
 ex <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c7">A1</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A2</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b1">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b1">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c8">A2</font> = <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c8">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span> &amp; <font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c8">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_3_1"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="lattices.html#T1" title="LATTICES:th.1">LATTICES:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T29" target="_self" title="LATTICE4:th.29">Th36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T9" target="_self" title="LATTICE4:th.9">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c8">A2</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T10" target="_self" title="LATTICE4:th.10">Th17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Maroon" title="c7">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c8">A2</font>,<span class="p2">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T41" target="_self" title="LATTICE4:th.41">Th48</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Maroon" title="c8">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:53_3_1_1"/>
<font color="Maroon" title="c7">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font></span><br/>

<span class="kw">thus </span><a NAME="E3:53_3_1_1"/>
<font color="Maroon" title="c8">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:53_3_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) = 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="lattice4.html#E24"><span class="lab"><font color="Green" title="E16">Lm1</font></span></a></span>
<br/>.= 
 <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">BL</font></span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice4.html#T10" target="_self" title="LATTICE4:th.10">Th17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:53_3"/>
( <font color="Maroon" title="c5">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b1">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b2">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b1">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b2">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Maroon" title="c5">B0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_3"><span class="lab"><font color="Green" title="E48">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:53"/>
 for <font color="Olive" title="b1">B</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="setwiseo.html#S4" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="#E6:53"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E46">A9</font></span></a>);<br/></span>
<a NAME="E8:53"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">B1</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice4.html#K1" title="LATTICE4:func.1">FinJoin</a> <font color="Olive" title="b2">A1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Olive" title="b3">A2</font>,<span class="p4">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>)</span>)</span></span>)</span> where <font color="Olive" title="b2">A1</font>, <font color="Olive" title="b3">A2</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> : ( <font color="Olive" title="b2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> &amp; <font color="Olive" title="b3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">AF</font> ) </span> } </span>  &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E45">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:53"/>
 ex <font color="Olive" title="b1">B0</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">BL</font> st <br/>( <font color="Olive" title="b1">B0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice4.html#K4" title="LATTICE4:func.4">SetImp</a> <font color="Maroon" title="c2">AF</font> &amp;  <a href="lattice2.html#K2" title="LATTICE2:func.2">FinJoin</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><a href="lattice4.html#K5" title="LATTICE4:func.5">comp</a> <font color="Maroon" title="c1">BL</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice4.html#K2" title="LATTICE4:func.2">FinMeet</a> <font color="Olive" title="b1">B0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
