reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;

theorem Th6:
  x in dom F & v = F.x implies (f (#) F).x = f.v * v
proof
  assume that
A1: x in dom F and
A2: v = F.x;
A3: F/.x = F.x by A1,PARTFUN1:def 6;
  len(f (#) F) = len F by Def5;
  then x in dom(f (#) F) by A1,FINSEQ_3:29;
  hence thesis by A2,A3,Def5;
end;
