reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th3:
  for a being Complex, x being complex-valued FinSequence holds
  len (a(#)x) = len x
proof
  let a be Complex, x be complex-valued FinSequence;
  set n=len x;
  x is FinSequence of COMPLEX by Lm2; then
  reconsider z=x as Element of n-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider a9 = a as Element of COMPLEX by XCMPLX_0:def 2;
  len (a9*z)=n by CARD_1:def 7;
  hence thesis;
end;
