reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;

theorem Th54:
  (for x being object st x in dom f holds f.x <= 0) implies f is nonpositive
proof
  assume
A1: for x being object st x in dom f holds f.x <= 0;
  let y be R_eal;
  assume y in rng f;
  then ex x being object st x in dom f & y = f.x by FUNCT_1:def 3;
  hence thesis by A1;
end;
