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 Th121:
  for x,y being real-valued FinSequence,a being Real st len x=
  len y holds |(a*x,y)| = a*|(x,y)|
proof
  let x,y be real-valued FinSequence,a be Real;
  reconsider a2=a as Element of REAL by XREAL_0:def 1;
A1:x is FinSequence of REAL & y is FinSequence of REAL by Lm2;
  assume len x=len y;
  then reconsider x2=x, y2 = y as Element of (len x)-tuples_on REAL by A1,
FINSEQ_2:92;
  |(a*x,y)| =Sum(a2*mlt(x2,y2)) by FINSEQOP:26
    .=a*|(x,y)| by Th87;
  hence thesis;
end;
