
theorem
  for T being complete LATTICE, S being full non empty SubRelStr of T
  st S is sups-inheriting holds S is complete
proof
  let T be complete LATTICE, S be full non empty SubRelStr of T;
  assume
A1: S is sups-inheriting;
  now
    let X be set;
    set Y = X /\ the carrier of S;
    reconsider Y as Subset of S by XBOOLE_1:17;
A2: ex_sup_of Y, T by YELLOW_0:17;
    then "\/"(Y,T) in the carrier of S by A1;
    then ex_sup_of Y, S by A2,YELLOW_0:64;
    hence ex_sup_of X, S by YELLOW_0:50;
  end;
  hence thesis by YELLOW_2:24;
end;
