reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th31:
  for S being sups-inheriting non empty full SubRelStr of L holds
  S is complete LATTICE
proof
  let S be sups-inheriting non empty full SubRelStr of L;
  for X being Subset of S holds ex_sup_of X, S
  proof
    let X be Subset of S;
A1: ex_sup_of X,L by YELLOW_0:17;
    then "\/"(X,L) in the carrier of S by YELLOW_0:def 19;
    hence thesis by A1,YELLOW_0:64;
  end;
  then S is complete by YELLOW_0:53;
  hence thesis;
end;
