reserve m, n for non zero Element of NAT;
reserve i, j, k for Element of NAT;
reserve Z for Subset of REAL 2;
reserve c for Real;
reserve I for non empty FinSequence of NAT;
reserve d1, d2 for Element of REAL;

theorem :: Th30Z:
  for u be Functional_Sequence of REAL 2, REAL, Z be Subset of REAL 2,
  c be Real st Z is open
  & for i be Nat holds (Z c= dom(u.i) & dom(u.i) = dom(u.0)
  & u.i is_partial_differentiable_on Z,<*1*> ^ <*1*>
  & u.i is_partial_differentiable_on Z,<*2*> ^ <*2*>
  & (for x, t be Real st <*x, t*> in Z holds
  (u.i)`partial|(Z, <*2*>^<*2*>)/.<*x, t*>
  = c^2*((u.i)`partial|(Z, <*1*>^<*1*>)/. <*x, t*>)))
  holds
  for i be Nat holds (Z c= dom(Partial_Sums(u).i)
  & Partial_Sums(u).i is_partial_differentiable_on Z,<*1*> ^ <*1*>
  & Partial_Sums(u).i is_partial_differentiable_on Z,<*2*> ^ <*2*>
  & (for x, t be Real st <*x, t*> in Z holds
  (Partial_Sums(u).i)`partial|(Z, <*2*>^<*2*>)/.<*x, t*>
  = c^2*((Partial_Sums(u).i)`partial|(Z, <*1*>^<*1*>)/.<*x, t*>)))
  proof
    let u be Functional_Sequence of REAL 2,REAL,
        Z be Subset of REAL 2, c be Real;
    assume that
    AS1: Z is open and
    AS2: for i be Nat holds (Z c= dom(u.i) & dom(u.i) = dom(u.0)
    & u.i is_partial_differentiable_on Z,<*1*> ^ <*1*>
    & u.i is_partial_differentiable_on Z,<*2*> ^ <*2*>
    & (for x, t be Real st <*x, t*> in Z holds
    (u.i)`partial|(Z, <*2*>^<*2*>)/.<*x, t*>
    = c^2*((u.i)`partial|(Z, <*1*>^<*1*>)/. <*x, t*>)));
    defpred X[Nat] means (Z c= dom(Partial_Sums(u).$1)
    & Partial_Sums(u).$1 is_partial_differentiable_on Z,<*1*> ^ <*1*>
    & Partial_Sums(u).$1 is_partial_differentiable_on Z,<*2*> ^ <*2*>
    & (for x, t be Real st <*x, t*> in Z holds
    (Partial_Sums(u).$1)`partial|(Z, <*2*>^<*2*>)/.<*x, t*>
    = c^2*((Partial_Sums(u).$1)`partial|(Z, <*1*>^<*1*>)/. <*x, t*>)));
    A9: for i being Nat st X[i] holds X[i+1]
    proof
      let i be Nat;
      set s = Partial_Sums(u).i;
      set v = u.(i+1);
      assume that
      A11: Z c= dom(s) and
      A12: s is_partial_differentiable_on Z,<*1*> ^ <*1*> and
      A13: s is_partial_differentiable_on Z,<*2*> ^ <*2*> and
      A14: for x, t be Real st <*x, t*> in Z holds
      s`partial|(Z, <*2*>^<*2*>)/.<*x, t*>
      = c^2*(s`partial|(Z, <*1*>^<*1*>)/. <*x, t*>);
      A15: Partial_Sums(u).(i+1) = s + v by MESFUN9C:def 2;
      Z c= dom(v)
      & dom v = dom(u.0)
      & v is_partial_differentiable_on Z,<*1*> ^ <*1*>
      & v is_partial_differentiable_on Z,<*2*> ^ <*2*>
      & (for x, t be Real st <*x, t*> in Z
      holds v`partial|(Z, <*2*>^<*2*>)/.<*x, t*>
      = c^2*(v`partial|(Z, <*1*>^<*1*>)/. <*x, t*>)) by AS2;
      hence thesis by Th30Y, A11, A12, A13, A14, AS1, A15;
    end;
    Partial_Sums(u).0 = u.0 by MESFUN9C:def 2; then
    A10: X[0] by AS2;
    for n being Nat holds X[n] from NAT_1:sch 2(A10, A9);
    hence thesis;
  end;
