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 x1,x2 being real-valued FinSequence st len x1=len x2 holds
  |(-x1, -x2)| = |(x1, x2)|
proof
  let x1,x2 be real-valued FinSequence;
  assume
A1: len x1=len x2;
  then len (-x2)=len x1 by Th114;
  then |(-x1, -x2)| = -|(x1, -x2)| by Th122
    .= --|(x1, x2)| by A1,Th122;
  hence thesis;
end;
