reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;

theorem Th9:
  f is_differentiable_in x0 implies r(#)f is_differentiable_in x0
  & diff((r(#)f),x0) = r*diff(f,x0)
proof
  assume f is_differentiable_in x0; then
  consider g be PartFunc of REAL,REAL-NS n such that
A1: f=g & g is_differentiable_in x0;
A2: r(#)g is_differentiable_in x0 & diff((r(#)g),x0) = r*diff(g,x0)
     by A1,NDIFF_3:16;
    reconsider r as Real;
A3: r(#)f = r(#)g by A1,NFCONT_4:6;
A4: diff(f,x0) = diff(g,x0) by A1,Th3;
   diff((r(#)f),x0) = diff((r(#)g),x0) by A1,Th3,NFCONT_4:6;
   hence thesis by A3,A2,A4,REAL_NS1:3;
end;
