theorem Th41:
  (FinMeet B)` = FinJoin (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).(M.(a,b))= J.((comp BL).a,(
  comp BL).b)
  proof
    let a,b be Element of BL;
    thus (comp BL).(M.(a,b))=(a"/\"b)` by Def12
      .=a`"\/"b` by LATTICES:23
      .= J.((comp BL).a,b`) by Def12
      .= J.((comp BL).a,(comp BL).b) by Def12;
  end;
A2: (comp BL).(the_unity_wrt M)= (the_unity_wrt M)` by Def12
    .=(Top BL )` by LATTICE2:57
    .=Bottom BL by Th29
    .= the_unity_wrt J by LATTICE2:52;
  thus (FinMeet B)`= (M$$(B,id BL))` by LATTICE2:def 4
    .= (comp BL).(M$$(B,id BL)) by Def12
    .= J$$(B,(comp BL)*(id BL)) by A2,A1,SETWISEO:36
    .= J$$(B, comp BL) by FUNCT_2:17
    .= FinJoin(B, comp BL) by LATTICE2:def 3;
end;
