reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
  for X,Y,Z be RealNormSpace,
          f be PartFunc of [:X,Y:],Z,
          U be Subset of [:X,Y:],
          I be LinearOperator of [:Y,X:],[:X,Y:]
   st U = dom f
    & f is_differentiable_on U
    & I is one-to-one onto isometric
    & ( for y be Point of Y,x be Point of X
        holds I.(y,x) = [x,y] )
  holds
    for a be Point of X, b be Point of Y,
        u be Point of [:X,Y:],
        v be Point of [:Y,X:]
     st u in U & u = [a,b] & v = [b,a]
    holds
      partdiff`1(f,u) = partdiff`2(f*I,v)
    & partdiff`2(f,u) = partdiff`1(f*I,v) by Th11;
