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 LM218:
  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`2 z holds
  r(#)f is_partial_differentiable_in`2 z &
  partdiff`2(r(#)f,z) = r*partdiff`2(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`2 z;
    D1:dom <*X,Y*> = Seg len <*X,Y*> by FINSEQ_1:def 3
    .= Seg 2 by FINSEQ_1:44;
    D3: 2 in dom <*X,Y*> by D1;
    then In(2,dom<*X,Y*>) = 2 by SUBSET_1:def 8; then
    BX2: <*X,Y*> .In(2,dom<*X,Y*>) = Y;
    P1: f*(IsoCPNrSP(X,Y)") is_partial_differentiable_in
    IsoCPNrSP(X,Y).z,2 by LM200,A1;
    P3: partdiff`2(r(#)f,z) = partdiff((r(#)f)*(IsoCPNrSP(X,Y)")
    ,IsoCPNrSP(X,Y).z,2) by LM210;
    P4: partdiff`2(f,z) = partdiff(f*(IsoCPNrSP(X,Y)"),IsoCPNrSP(X,Y).z,2)
    by LM210;
    P6: (r(#)f)*(IsoCPNrSP(X,Y)") = r(#)(f*(IsoCPNrSP(X,Y)")) by Th27;
    r(#)(f*(IsoCPNrSP(X,Y)")) is_partial_differentiable_in IsoCPNrSP(X,Y).z,2
    by NDIFF_5:30,P1,D3; then
    (r(#)f)*(IsoCPNrSP(X,Y)") is_partial_differentiable_in IsoCPNrSP(X,Y).z,2
    by Th27;
    hence (r(#)f) is_partial_differentiable_in`2 z by LM200;
    thus thesis by BX2,D3,P1,P3,P4,P6,NDIFF_5:30;
  end;
