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 Th3:
  for p being FinSequence st k in dom p & not k + 1 in dom p holds len p = k
proof
  let p be FinSequence such that
A1: k in dom p and
A2: not k + 1 in dom p;
A3: 1 > k + 1 or k + 1 > len p by A2,FINSEQ_3:25;
A4: 1 + 0 <= k + 1 by XREAL_1:7;
  k <= len p by A1,FINSEQ_3:25;
  hence thesis by A3,A4,NAT_1:22;
end;
