
theorem Th12:
  for X be Group-Sequence, x,y,z be Element of product X,
  x1,y1,z1 be FinSequence st x = x1 & y = y1 & z = z1 holds
  z = x+y iff for j be Element of dom carr X holds
  z1.j = (the addF of (X.j)).((x1.j),(y1.j))
proof
  let X be Group-Sequence, x,y,z be Element of product X, x1,y1,z1 be
  FinSequence;
  assume
A1: x = x1 & y = y1 & z = z1;
  hereby assume
A2: z = x+y;
    thus for j be Element of dom carr X holds z1.j = (the addF of (X.j)).
    ((x1.j),(y1.j))
    proof
      let j be Element of dom carr X;
      thus z1.j = ((addop X).j).((x1.j),(y1.j)) by A1,A2,PRVECT_1:def 8
        .= (the addF of (X.j)).((x1.j),(y1.j)) by PRVECT_1:def 12;
    end;
  end;
  assume
A3: for j be Element of dom carr X holds z1.j = (the addF of (X.j)).
  ((x1.j),(y1.j));
  reconsider Ixy = x+y as FinSequence by NDIFF_5:9;
A4: dom Ixy = dom carr X by CARD_3:9;
  for j0 being Nat st j0 in dom z1 holds z1.j0 = Ixy.j0
  proof
    let j0 be Nat;
    assume j0 in dom z1;
    then reconsider j = j0 as Element of dom carr X by CARD_3:9,A1;
    Ixy. j0 = ((addop X).j).((x1.j),(y1.j)) by A1,PRVECT_1:def 8
      .= (the addF of (X.j)).((x1.j),(y1.j)) by PRVECT_1:def 12
      .= z1.j by A3;
    hence thesis;
  end;
  hence z = x + y by A1,CARD_3:9,A4,FINSEQ_1:13;
end;
