
theorem Th15:
  for X be non empty set, Y be set, f be PartFunc of X,ExtREAL st
  f is nonnegative holds f|Y is nonnegative
proof
  let X be non empty set, Y be set, f be PartFunc of X,ExtREAL;
  assume
A1: f is nonnegative;
  now
    let x be object;
    assume
A2: x in dom(f|Y);
    then (f|Y).x = f.x by FUNCT_1:47;
    hence 0 <= (f|Y).x by A1,A2,SUPINF_2:39;
  end;
  hence thesis by SUPINF_2:52;
end;
