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 Th116:
  for x1,x2 being real-valued FinSequence st len x1=len x2 holds
  len (x1-x2)=len x1
proof
  let x1,x2 be real-valued FinSequence;
  set n=len x1;
A1: x2 is FinSequence of REAL by Lm2;
  x1 is FinSequence of REAL by Lm2;
  then reconsider z1=x1 as Element of (len x1)-tuples_on REAL by FINSEQ_2:92;
  assume len x1=len x2;
  then reconsider z2=x2 as Element of n-tuples_on REAL by A1,FINSEQ_2:92;
  dom (z1-z2)=Seg n by FINSEQ_2:124;
  hence thesis by FINSEQ_1:def 3;
end;
