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 Th58:
  (for x be set st x in dom f /\ dom g holds g.x <= f.x) implies f
  -g is nonnegative
proof
  assume
A1: for x be set st x in dom f /\ dom g holds g.x <= f.x;
  now
    let x be object such that
A2: x in dom(f-g);
    dom(f-g) = dom f /\ dom g by VALUED_1:12;
    then 0 <= f.x - g.x by A1,A2,XREAL_1:48;
    hence 0 <= (f-g).x by A2,VALUED_1:13;
  end;
  hence thesis by Th52;
end;
