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 Th52:
  X c= the_transitive-closure_of X
proof
  let x be object such that
A1: x in X;
  consider f such that
A2: dom f = omega and
A3: f.0 = X and
A4: for n being Nat holds f.(succ n) = f(n,f.n)
        from ORDINAL2:sch 18;
   reconsider z = 0 as Element of omega by ORDINAL1:def 12;
    x in f.z by A1,A3;
  hence x in the_transitive-closure_of X by A2,A3,A4,Def7;
end;
