reserve x, y, X for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u1, v, v1, v2, w, w9, w1, w2 for Element of E^omega;
reserve F for Subset of E^omega;
reserve i, k, l for Nat;
reserve TS for non empty transition-system over F;
reserve S, T for Subset of TS;

theorem Th3:
  for p, q being FinSequence st k in dom p & len p + 1 = len q
  holds k + 1 in dom q
proof
  let p, q be FinSequence such that
A1: k in dom p and
A2: len p + 1 = len q;
  k <= len p by A1,FINSEQ_3:25;
  then 1 + 0 <= k + 1 & k + 1 <= len p + 1 by XREAL_1:7;
  hence thesis by A2,FINSEQ_3:25;
end;
