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 LM43:
  for A, B, C, d, c, e be Real, f, g be Function of REAL,REAL st
  (for x be Real holds f.x = A*cos.(e*x) + B*sin.(e*x)) &
  (for t be Real holds g.t = C*cos.((e*c)*t) + d*sin.((e*c)*t))
  holds
  for x, t be Real holds
  f/.x*((diff(g,[#]REAL).2)/.t) = c^2*(diff(f,[#]REAL).2)/.x *g/.t
  proof
    let A, B, C, d, c, e be Real, f, g be Function of REAL,REAL;
    assume
    AS: (for x be Real holds f.x = A*cos.(e*x) + B*sin.(e*x)) &
    (for t be Real holds g.t = C*cos.((e*c)*t) + d*sin.((e*c)*t));
    let x, t be Real;
    (diff(f,[#]REAL).2)/.x + e^2*f/.x = 0 by AS, LM41; then
    Q3: c^2*(diff(f,[#]REAL).2)/.x *g/.t = c^2 *(-e^2*f/.x)*g/.t
    .= - c^2 *e^2*f/.x*g/.t;
    (diff(g,[#]REAL).2)/.t + (e*c)^2*g/.t = 0 by AS, LM41;
    then f/.x*((diff(g,[#]REAL).2)/.t) = f/.x*(-(e*c)^2*g/.t)
    .=f/.x*(-((e*c)*(e*c))*g/.t) by SQUARE_1:def 1
    .=f/.x*(-((e*e)*(c*c))*g/.t)
    .=f/.x*(-(e^2*(c*c))*g/.t) by SQUARE_1:def 1
    .=f/.x*(-(e^2*c^2)*g/.t) by SQUARE_1:def 1
    .= - c^2 *e^2*f/.x*g/.t;
    hence thesis by Q3;
  end;
