
theorem Th7:
  for X being non empty set for G being Function of X,ExtREAL st G
  is nonnegative holds On G is nonnegative
proof
  let X be non empty set;
  let G be Function of X,ExtREAL;
  assume
A1: G is nonnegative;
  for n being Element of NAT holds 0. <= (On G).n
  proof
    let n be Element of NAT;
    per cases;
    suppose
A2:   n in X;
      then (On G).n = G.n by Def1;
      hence thesis by A1,A2,SUPINF_2:39;
    end;
    suppose
      not n in X;
      hence thesis by Def1;
    end;
  end;
  hence thesis by SUPINF_2:39;
end;
