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 Th124:
  for x1,x2,x3 being real-valued FinSequence st len x1=len x2 & len x2
  =len x3 holds |(x1-x2, x3)| = |(x1, x3)| - |(x2, x3)|
proof
  let x1,x2,x3 be real-valued FinSequence;
  assume that
A1: len x1=len x2 and
A2: len x2=len x3;
  len (-x2)=len x2 by Th114;
  then |(x1 - x2, x3)| = |(x1, x3)| + |(-x2, x3)| by A1,A2,Th120
    .= |(x1, x3)| + - |(x2, x3)| by A2,Th122;
  hence thesis;
end;
