reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem
  for X, A being set, f being Function of X, REAL, s being Real holds
    (s+f).:A = s++(f.:A)
proof
  let X, A be set, f be Function of X, REAL, s be Real;
  now
    let x be object;
    hereby
      assume x in (s+f).:A;
      then consider x1 being object such that
A1:   x1 in X & x1 in A & x = (s+f).x1 by FUNCT_2:64;
      x = s+(f.x1) & f.x1 in f.:A by A1,FUNCT_2:35,VALUED_1:2; then
      x in { s + q3 : q3 in f.:A };
      hence x in s++(f.:A) by Lm5;
    end;
    assume x in s++(f.:A); then
    x in { s + q3 : q3 in f.:A } by Lm5;
    then consider r3 such that
A2: x = s+r3 and
A3: r3 in f.:A;
    consider x1 being object such that
A4: x1 in X and
A5: x1 in A and
A6: r3 = f.x1 by A3,FUNCT_2:64;
    x = (s+f).x1 by A2,A4,A6,VALUED_1:2;
    hence x in (s+f).:A by A4,A5,FUNCT_2:35;
  end;
  hence thesis by TARSKI:2;
end;
