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
  for x,y being Real,x1,x2,x3 being real-valued FinSequence st len x1
  =len x2 & len x2=len x3 holds |((x*x1+y*x2), x3)| = x*|(x1,x3)| + y*|(x2,x3)|
proof
  let x,y be Real,x1,x2,x3 be real-valued FinSequence;
  assume that
A1: len x1=len x2 and
A2: len x2=len x3;
  len (x*x1)=len x1 & len (y*x2)=len x2 by Th117;
  then |(x*x1 + y*x2, x3)| = |(x*x1, x3)| + |(y*x2, x3)| by A1,A2,Th120
    .= x*|(x1,x3)| + |(y*x2,x3)| by A1,A2,Th121
    .= x*|(x1,x3)| + y*|(x2,x3)| by A2,Th121;
  hence thesis;
end;
