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 LM200:
  for X, Y, W be RealNormSpace,
  z be Point of [:X,Y:],
  f be PartFunc of [:X,Y:], W holds
  (f is_partial_differentiable_in`1 z iff
  f*(IsoCPNrSP(X,Y)") is_partial_differentiable_in IsoCPNrSP(X,Y).z,1 ) &
  (f is_partial_differentiable_in`2 z iff
  f*(IsoCPNrSP(X,Y)") is_partial_differentiable_in IsoCPNrSP(X,Y).z,2 )
  proof
    let X, Y, W be RealNormSpace,
    z be Point of [:X,Y:],
    f be PartFunc of [:X,Y:], W;
    reconsider g = f*(IsoCPNrSP(X,Y)") as PartFunc of product <*X,Y*>, W;
    reconsider w = IsoCPNrSP(X,Y).z as Element of product <*X,Y*>;
    D1:dom <*X,Y*> = Seg len <*X,Y*> by FINSEQ_1:def 3
    .= Seg 2 by FINSEQ_1:44;
    then 1 in dom <*X,Y*>; then
    AS1: In(1,dom<*X,Y*>) = 1 by SUBSET_1:def 8;
    2 in dom <*X,Y*> by D1; then
    AS2:In(2,dom<*X,Y*>) = 2 by SUBSET_1:def 8;
    X1: f*reproj1(z) = f*((IsoCPNrSP(X,Y)")
    *reproj(In(1,dom<*X,Y*>),IsoCPNrSP(X,Y).z)) by LM180
    .= g * reproj(In(1,dom<*X,Y*>),w) by RELAT_1:36;
    X3: X = <*X,Y*>. In(1,dom<*X,Y*>) by AS1;
    Y1: f*reproj2(z) = f*((IsoCPNrSP(X,Y)")
    *reproj(In(2,dom<*X,Y*>),IsoCPNrSP(X,Y).z)) by LM180
    .= g * reproj(In(2,dom<*X,Y*>),w) by RELAT_1:36;
    Y3: Y = <*X,Y*>. In(2,dom<*X,Y*>) by AS2;
    thus f is_partial_differentiable_in`1 z iff
    f*(IsoCPNrSP(X,Y)") is_partial_differentiable_in IsoCPNrSP(X,Y).z,1
    by X1,X3,LM190;
    thus thesis by Y1,Y3,LM190;
  end;
