let BL be Boolean Lattice; :: thesis: for B being Element of Fin the carrier of BL holds (FinJoin B) ` = FinMeet (B,(comp BL))
let B be Element of Fin 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;
A1: 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 Def12
.= (a `) "/\" (b `) by LATTICES:24
.= the L_meet of BL . (((comp BL) . a),(b `)) by Def12
.= the L_meet of BL . (((comp BL) . a),((comp BL) . b)) by Def12 ; :: thesis: verum

end;

A2: (comp BL) . (the_unity_wrt the L_join of BL) = (the_unity_wrt the L_join of BL) ` by Def12
.= (Bottom BL) ` by LATTICE2:52
.= Top BL by Th30
.= the_unity_wrt the L_meet of BL by LATTICE2:57 ;
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 Def12
.= the L_meet of BL $$ (B,((comp BL) * (id BL))) by A2, A1, SETWISEO:36
.= the L_meet of BL $$ (B,(comp BL)) by FUNCT_2:17
.= FinMeet (B,(comp BL)) by LATTICE2:def 4 ; :: thesis: verum