reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;

theorem
  w1^v1 = w2^v2 & ( len w1 = len w2 or len v1 = len v2 ) implies w1 = w2
  & v1 = v2
proof
  assume that
A1: w1^v1 = w2^v2 and
A2: len w1 = len w2 or len v1 = len v2;
A3: len w1 + len v1 = len (w2^v2) by A1,AFINSQ_1:17
    .= len w2 + len v2 by AFINSQ_1:17;
  now
    let k be Nat;
    assume
A4: k in dom w1;
    hence w1.k = (w2^v2).k by A1,AFINSQ_1:def 3
      .= w2.k by A2,A3,A4,AFINSQ_1:def 3;
  end;
  hence w1 = w2 by A2,A3,AFINSQ_1:8;
  hence thesis by A1,AFINSQ_1:28;
end;
