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

theorem Th37:
  for D be non empty set, F be PartFunc of D,REAL, d be Element of
  D holds 0<=(max+ F).d
proof
  let D be non empty set, F be PartFunc of D,REAL, d be Element of D;
A1: dom F = dom (max+ F) by Def10;
  per cases;
  suppose
    d in dom F;
    then (max+ F).d = max+(F.d) by A1,Def10
      .= max(F.d,0);
    hence thesis by XXREAL_0:25;
  end;
  suppose
    not d in dom F;
    hence thesis by A1,FUNCT_1:def 2;
  end;
end;
