let BL be Boolean Lattice; :: thesis: for B being Finite_Subset of the carrier of BL holds (FinJoin B) ` = FinMeet (B,(comp BL))
let B be Finite_Subset of the carrier of BL; :: thesis: (FinJoin B) ` = FinMeet (B,(comp BL))
set M = the L_meet of BL;
set J = the L_join of BL;
A3: for a, b being Element of BL holds (comp BL) . ( the L_join of BL . (a,b)) = the L_meet of BL . (((comp BL) . a),((comp BL) . b))

proof

let a, b be Element of BL; :: thesis: (comp BL) . ( the L_join of BL . (a,b)) = the L_meet of BL . (((comp BL) . a),((comp BL) . b))
thus (comp BL) . ( the L_join of BL . (a,b)) = (a "\/" b) ` by Def17
.= (a `) "/\" (b `) by LATTICES:51
.= the L_meet of BL . (((comp BL) . a),(b `)) by Def17
.= the L_meet of BL . (((comp BL) . a),((comp BL) . b)) by Def17 ; :: thesis: verum

end;

A4: (comp BL) . (the_unity_wrt the L_join of BL) = (the_unity_wrt the L_join of BL) ` by Def17
.= (Bottom BL) ` by LATTICE2:68
.= Top BL by Th37
.= the_unity_wrt the L_meet of BL by LATTICE2:74 ;
thus (FinJoin B) ` = ( the L_join of BL $$ (B,(id BL))) ` by LATTICE2:def 3
.= (comp BL) . ( the L_join of BL $$ (B,(id BL))) by Def17
.= the L_meet of BL $$ (B,((comp BL) * (id BL))) by A4, A3, SETWISEO:45
.= the L_meet of BL $$ (B,(comp BL)) by FUNCT_2:23
.= FinMeet (B,(comp BL)) by LATTICE2:def 4 ; :: thesis: verum