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 Th56:
  f is nonnegative & g is nonnegative implies f+g is nonnegative
proof
  assume that
A1: f is nonnegative and
A2: g is nonnegative;
  for x be object st x in dom(f+g) holds 0 <= (f+g).x
  proof
    let x be object such that
A3: x in dom(f+g);
    0 <= f.x by A1,Th51;
    then
A4: g.x <= f.x + g.x by XREAL_1:31;
    0 <= g.x by A2,Th51;
    hence thesis by A3,A4,VALUED_1:def 1;
  end;
  hence thesis by Th52;
end;
