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 Th4:
  len u = 1 implies ex e st <%e%> = u & e = u.0
proof
  assume len u = 1;
  then len u = 0 + 1;
  then consider v, e such that
A1: len v = 0 and
A2: u = v^<%e%> by FLANG_1:7;
  take e;
 v = <%>E by A1;
  then u = {}^<%e%> by A2;
  then u = <%e%>;
  hence thesis;
end;
