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;

theorem
  X is epsilon-transitive & Y is epsilon-transitive implies
  X /\ Y is epsilon-transitive
proof
  assume that
A1: Z in X implies Z c= X and
A2: Z in Y implies Z c= Y;
  let Z;
  assume
A3: Z in X /\ Y;
then A4: Z in X by XBOOLE_0:def 4;
A5: Z in Y by A3,XBOOLE_0:def 4;
A6: Z c= X by A1,A4;
 Z c= Y by A2,A5;
  hence thesis by A6,XBOOLE_1:19;
end;
