reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th24:
  i in dom F & v = F.i implies (f (#) F).i = f.v * v
proof
  assume that
A1: i in dom F and
A2: v = F.i;
A3: F/.i = F.i by A1,PARTFUN1:def 6;
  len(f (#) F) = len F by Def7;
  then i in dom(f (#) F) by A1,FINSEQ_3:29;
  hence thesis by A2,A3,Def7;
end;
