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 LM40:
  for f, g be Function of REAL,REAL, u be PartFunc of REAL 2,REAL,
  c be Real st f is_differentiable_on 2,[#]REAL &
  g is_differentiable_on 2,[#]REAL & dom u = [#](REAL 2) &
  (for x, t be Real holds u/.<*x, t*> = f/.x*g/.t) &
  for x, t be Real holds
  f/.x*((diff(g,[#]REAL).2)/.t) = c^2*(diff(f,[#]REAL).2)/.x*g/.t
  holds
  u is_partial_differentiable_on [#](REAL 2),<*1*> ^ <*1*> &
  (for x, t be Real st x in [#]REAL & t in [#]REAL holds
  u `partial| ([#](REAL 2),<*1*> ^ <*1*>)/.<*x, t*>
  = (diff(f,[#]REAL).2)/.x*g/.t) &
  u is_partial_differentiable_on [#](REAL 2),<*2*> ^ <*2*> &
  (for x, t be Real st x in [#]REAL & t in [#]REAL holds
  u `partial| ([#](REAL 2),<*2*>^<*2*>)/.<*x, t*>
  = f/.x*((diff(g,[#]REAL).2)/.t)) &
  for x, t be Real holds
  u`partial|([#](REAL 2), <*2*>^<*2*>)/.<*x, t*>
  = c^2*(u`partial|([#](REAL 2), <*1*>^<*1*>)/.<*x, t*>)
  proof
    let f, g be Function of REAL,REAL, u be PartFunc of REAL 2,REAL,
    c be Real;
    assume that
    AS1: f is_differentiable_on 2,[#]REAL and
    AS2: g is_differentiable_on 2,[#]REAL and
    AS3: dom u = [#](REAL 2) and
    AS4: for x, t be Real holds u/.<*x, t*> = f/.x*g/.t and
    AS5: for x, t be Real holds
    f/.x*((diff(g,[#]REAL).2)/.t) = c^2*(diff(f,[#]REAL).2)/.x*g/.t;
    P1: [#]REAL = dom f & [#]REAL = dom g by FUNCT_2:def 1;
    P4: for x, t be Real st x in dom f & t in dom g holds
    u/.<*x, t*> = f/.x*g/.t by AS4;
    for x, t be Real holds
    u`partial|([#](REAL 2), <*2*>^<*2*>)/.<*x, t*>
    = c^2*(u`partial|([#](REAL 2), <*1*>^<*1*>)/.<*x, t*>)
    proof
      let x, t be Real;
      X1: f/.x*((diff(g,[#]REAL).2)/.t)
      = c^2*(diff(f,[#]REAL).2)/.x*g/.t by AS5;
      X3: x in [#]REAL & t in [#]REAL by XREAL_0:def 1; then
      u `partial| ([#](REAL 2),<*1*> ^ <*1*>)/.<*x, t*>
      = (diff(f,[#]REAL).2)/.x *g/.t
      by LM30, AS1, AS2, AS3, P1, LMOP3, P4;
      hence thesis
      by LM30, AS1, AS2, AS3, P1, LMOP3, P4, X1, X3;
    end;
    hence thesis by LM30, AS1, AS2, AS3, P1, LMOP3, P4;
  end;
