reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;
reserve n for Element of omega;

theorem Th54:
  (for Z st X c= Z & Z is epsilon-transitive holds Y c= Z) & X c= Y &
  Y is epsilon-transitive implies the_transitive-closure_of X = Y
by Th53,Th52;
