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;
