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 Th127:
  for x1,x2,y1,y2 being real-valued FinSequence st len x1=len x2 & len
x2=len y1 & len y1=len y2 holds |(x1-x2, y1-y2)| = |(x1, y1)| - |(x1, y2)| - |(
  x2, y1)| + |(x2, y2)|
proof
  let x1,x2,y1,y2 be real-valued FinSequence;
  assume that
A1: len x1=len x2 and
A2: len x2=len y1 and
A3: len y1=len y2;
  |(x1,y1-y2)| = |(x1,y1)| - |(x1,y2)| by A1,A2,A3,Th124;
  then
A4: |(x1,y1-y2)| - |(x2,y1-y2)| = (|(x1,y1)|-|(x1,y2)|)-(|(x2,y1)|-|(x2,y2)|
  ) by A2,A3,Th124;
  len (y1 - y2)=len y1 by A3,Th116;
  hence thesis by A1,A2,A4,Th124;
end;
