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

theorem Th32: ::Remark 2.4 (Part I, variant 1)
  for M being non empty RelStr for f being Function of L, M st f
  is sups-preserving holds Image f is sups-inheriting
proof
  let M be non empty RelStr;
  let f be Function of L, M such that
A1: f is sups-preserving;
  set S = subrelstr(rng f);
  for Y being Subset of S st ex_sup_of Y,M holds "\/"(Y, M) in the carrier of S
  proof
    let Y be Subset of S;
    assume ex_sup_of Y,M;
A2: f preserves_sup_of (f"Y) & ex_sup_of f"Y,L by A1,YELLOW_0:17;
    Y c= the carrier of S;
    then Y c= rng f by YELLOW_0:def 15;
    then "\/"(Y, M) = sup(f.:(f"Y)) by FUNCT_1:77
      .= f.sup(f"Y) by A2;
    then "\/"(Y, M) in rng f by FUNCT_2:4;
    hence thesis by YELLOW_0:def 15;
  end;
  hence thesis by YELLOW_0:def 19;
end;
