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

theorem Th30:
  for S being infs-inheriting non empty full SubRelStr of L holds
  S is complete LATTICE
proof
  let S be infs-inheriting non empty full SubRelStr of L;
  for X being Subset of S holds ex_inf_of X,S
  proof
    let X be Subset of S;
A1: ex_inf_of X,L by YELLOW_0:17;
    then "/\"(X,L) in the carrier of S by YELLOW_0:def 18;
    hence thesis by A1,YELLOW_0:63;
  end;
  then S is complete by Th28;
  hence thesis;
end;
