reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th46:
  for D be non empty set, F be PartFunc of D,REAL st (for d be
  Element of D st d in dom F holds F.d>=0) holds max+ F = F
proof
  let D be non empty set, F be PartFunc of D,REAL;
A1: dom(max+ F) = dom F by Def10;
  assume
A2: for d be Element of D st d in dom F holds F.d>=0;
  now
    let d be Element of D;
    assume
A3: d in dom F;
    then
A4: F.d>=0 by A2;
    thus (max+ F).d = max+(F.d) by A1,A3,Def10
      .= F.d by A4,XXREAL_0:def 10;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
