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 Th120:
  for x,y,z being real-valued FinSequence st len x=len y & len y=len z
  holds |((x+y),z)| = |(x,z)| + |(y,z)|
proof
  let x,y,z be real-valued FinSequence;
A1:x is FinSequence of REAL & y is FinSequence of REAL &
  z is FinSequence of REAL by Lm2;
  assume
A2: len x=len y & len y=len z;
  then reconsider x2=x,y2=y,z2=z as Element of (len x)-tuples_on REAL by A1,
FINSEQ_2:92;
  |((x+y),z)|= Sum(mlt(x,z)+mlt(y,z)) by A2,Th118
    .= Sum(mlt(x2,z2))+Sum(mlt(y2,z2)) by Th89
    .= |(x,z)| + |(y,z)|;
  hence thesis;
end;
