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 Th79:
  not <%>E in rng dom (the Tran of TS) implies (==>.-relation(TS)
reduces [x, <%e%>], [y, <%>E] implies [[x, <%e%>], [y, <%>E]] in ==>.-relation(
  TS))
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
  assume ==>.-relation(TS) reduces [x, <%e%>], [y, <%>E];
  then consider P being RedSequence of ==>.-relation(TS) such that
A2: P.1 = [x, <%e%>] & P.len P = [y, <%>E] by REWRITE1:def 3;
A3: len P = 1 + 1 by A1,A2,Th63;
  then 1 in dom P & 1 + 1 in dom P by FINSEQ_3:25;
  hence thesis by A2,A3,REWRITE1:def 2;
end;
