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 Th24:
  for x,y,z being complex-valued FinSequence st len x=len y & len y=len z
  holds x + (y + z) = x + y + z
proof
  let x,y,z be complex-valued FinSequence;
  reconsider x1=x, y1 = y, z1 = z as FinSequence of COMPLEX by Lm2;
  assume
A1: len x=len y & len y=len z;
  reconsider z9=z1 as Element of (len z)-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider y9=y1 as Element of (len y)-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider x9=x1 as Element of (len x)-tuples_on COMPLEX by FINSEQ_2:92;
  x + (y + z) = addcomplex.:(x1,y1+z1) by SEQ_4:def 6
    .= addcomplex.:(x1,addcomplex.:(y1,z1)) by SEQ_4:def 6
    .= addcomplex.:(addcomplex.:(x9,y9),z9) by A1,FINSEQOP:28
    .= addcomplex.:(x1+y1,z1) by SEQ_4:def 6
    .= x + y + z by SEQ_4:def 6;
  hence thesis;
