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;
