reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th26:
  for x,y being complex-valued FinSequence st len x=len y & x + y = 0c
  (len x) holds x = -y & y = -x
proof
  let x,y be complex-valued FinSequence;
  assume that
A1: len x=len y and
A2: x + y = 0c (len x);
  reconsider x1=x, y1 = y as FinSequence of COMPLEX by Lm2;
  reconsider x9=x1 as Element of (len x)-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider y9=y1 as Element of (len y)-tuples_on COMPLEX by FINSEQ_2:92;
A3: x + y = addcomplex.:(x1,y1) & -y = compcomplex*y1 by SEQ_4:def 6,def 8;
  x + y = (len x) |-> 0c by A2,SEQ_4:def 12;
  then x9 = -y9 by A1,A3,BINOP_2:1,FINSEQOP:74,SEQ_4:51,52;
  hence thesis;
end;
