reserve x,y,X,X1,Y,Z for set;
reserve L for Lattice;
reserve F,H for Filter of L;
reserve p,q,r for Element of L;
reserve L1, L2 for Lattice;
reserve a1,b1 for Element of L1;
reserve a2 for Element of L2;
reserve f for Homomorphism of L1,L2;
reserve B for Element of Fin the carrier of L;
reserve DL for distributive Lattice;
reserve f for Homomorphism of DL,L2;
reserve 0L for lower-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 0L,
  b for Element of 0L;
reserve f for UnOp of the carrier of 0L;
reserve 1L for upper-bounded Lattice,
  B,B1,B2 for Element of Fin the carrier of 1L,
  b for Element of 1L;
reserve f,g for UnOp of the carrier of 1L;
reserve DL for distributive upper-bounded Lattice,
  B for Element of Fin the carrier of DL,
  p for Element of DL,
  f for UnOp of the carrier of DL;
reserve CL for C_Lattice;
reserve IL for implicative Lattice;
reserve f for Homomorphism of IL,CL;
reserve i,j,k for Element of IL;
reserve BL for Boolean Lattice;
reserve f for Homomorphism of BL,CL;
reserve A for non empty Subset of BL;
reserve a1,a,b,c,p,q for Element of BL;
reserve B,B0,B1,B2,A1,A2 for Element of Fin the carrier of BL;
reserve F,H for Field of BL;

theorem
  (FinJoin B)` = FinMeet (B,comp BL)
proof
  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).(J.(a,b))= M.((comp BL).a,(
  comp BL).b)
  proof
    let a,b be Element of BL;
    thus (comp BL).(J.(a,b))=(a"\/"b)` by Def12
      .=a`"/\"b` by LATTICES:24
      .= M.((comp BL).a,b`) by Def12
      .= M.((comp BL).a,(comp BL).b) by Def12;
  end;
A2: (comp BL).(the_unity_wrt J)= (the_unity_wrt J)` by Def12
    .=(Bottom BL )` by LATTICE2:52
    .=Top BL by Th30
    .= the_unity_wrt M by LATTICE2:57;
  thus (FinJoin B)`= (J$$(B,id BL))` by LATTICE2:def 3
    .= (comp BL).(J$$(B,id BL)) by Def12
    .= M$$(B,(comp BL)*(id BL)) by A2,A1,SETWISEO:36
    .= M$$(B, comp BL) by FUNCT_2:17
    .= FinMeet(B, comp BL) by LATTICE2:def 4;
end;
