reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem Th12:
  ((0*n)+*(x,c))^2 = (0*n)+*(x,c^2)
  proof
    set f = (0*n)+*(x,c);
    set g = (0*n)+*(x,c^2);
A1: dom f = dom (0*n) by FUNCT_7:30;
A2: dom g = dom (0*n) by FUNCT_7:30;
A3: dom (f^2) = dom f by VALUED_1:11;
    thus dom (f^2) = dom g by A1,A2,VALUED_1:11;
    let a be object;
    assume
A4: a in dom (f^2);
A5: (f^2).a = (f.a)^2 by VALUED_1:11;
    per cases;
    suppose
A6:   a = x;
      then f.a = c by A1,A3,A4,FUNCT_7:31;
      hence thesis by A6,A1,A3,A4,A5,FUNCT_7:31;
    end;
    suppose
A7:   a <> x;
      then
A8:   f.a = (n|->0).a by FUNCT_7:32
      .= {}.x;
      g.a = (n|->0).a by A7,FUNCT_7:32
      .= {}.x;
      hence thesis by A5,A8;
    end;
  end;
