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 LM217:
  for X, Y, W be RealNormSpace,
  z be Point of [:X,Y:],
  r be Real,
  f be PartFunc of [:X,Y:], W
  st f is_partial_differentiable_in`1 z holds
  r(#)f is_partial_differentiable_in`1 z &
  partdiff`1(r(#)f,z) = r*partdiff`1(f,z)
  proof
    let X, Y, W be RealNormSpace,
    z be Point of [:X,Y:],
    r be Real,
    f be PartFunc of [:X,Y:], W;
    assume A1: f is_partial_differentiable_in`1 z;
    dom <*X,Y*> = Seg len <*X,Y*> by FINSEQ_1:def 3
    .= Seg 2 by FINSEQ_1:44; then
    D2:1 in dom <*X,Y*>;
    then In(1,dom<*X,Y*>) = 1 by SUBSET_1:def 8; then
    BX1: <*X,Y*> .In(1,dom<*X,Y*>) = X;
    P1: f*(IsoCPNrSP(X,Y)") is_partial_differentiable_in
    IsoCPNrSP(X,Y).z,1 by LM200,A1;
    P3: partdiff`1(r(#)f,z) = partdiff((r(#)f)*(IsoCPNrSP(X,Y)")
    ,IsoCPNrSP(X,Y).z,1) by LM210;
    P4: partdiff`1(f,z) = partdiff(f*(IsoCPNrSP(X,Y)"),IsoCPNrSP(X,Y).z,1)
    by LM210;
    P6: (r(#)f)*(IsoCPNrSP(X,Y)") = r(#)(f*(IsoCPNrSP(X,Y)")) by Th27;
    hence (r(#)f) is_partial_differentiable_in`1 z
    by D2,P1,LM200,NDIFF_5:30;
    thus thesis by BX1,D2,P1,P3,P4,P6,NDIFF_5:30;
  end;
