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;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem
  not <%>E in rng dom (the Tran of TS) implies <%>E-succ_of (S, TS) = S
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
A2: now
    let x be object;
    assume x in <%>E-succ_of (S, TS);
    then ex s st s in S & s, <%>E ==>* x, TS by Th103;
    hence x in S by A1,Th100;
  end;
  now
    let x be object;
    assume
A3: x in S;
    x, <%>E ==>* x, TS by Th95;
    hence x in <%>E-succ_of (S, TS) by A3;
  end;
  hence thesis by A2,TARSKI:2;
end;
