reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;
reserve a,b for Aleph;
reserve a,b for Aleph;

theorem Th30:
  a c= b implies exp(a,b) = exp(2,b)
proof
A1: card bool a = exp(2,card a) by CARD_2:31;
  card a = a & card a in card bool a by CARD_1:14;
  then
A2: exp(a,b) c= exp(exp(2,a),b) by A1,CARD_2:92;
  assume a c= b;
  then
A3: exp(exp(2,a),b) = exp(2,a*`b) & a*`b = b by Th17,CARD_2:30;
  2 in a by Lm2,Th15;
  then exp(2,b) c= exp(a,b) by CARD_2:92;
  hence thesis by A2,A3;
end;
