reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;
reserve z,z1,z2 for Element of COMPLEX;
reserve n for Nat,
  x, y, a for Real,
  p, p1, p2, p3, q, q1, q2 for Element of n-tuples_on REAL;

theorem Th117:
  for a being Real, x being real-valued FinSequence
   holds len (a*x)=len x
proof
  let a be Real, x be real-valued FinSequence;
  set n=len x;
  x is FinSequence of REAL by Lm2;
  then reconsider z=x as Element of n-tuples_on REAL by FINSEQ_2:92;
  len (a*z)=n by CARD_1:def 7;
  hence thesis;
end;
