reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;

theorem Th25:
  for X,Y being ordinal-membered set
  st for x,y st x in X & y in Y holds x in y
  holds numbering(X \/ Y) = (numbering X)^(numbering Y)
  proof
    let X,Y be ordinal-membered set; assume
A1: for x,y st x in X & y in Y holds x in y;
    set f = numbering X, g = numbering Y, h = numbering(X\/Y);
    set a = ord-type X, b = ord-type Y;
    set P = RelIncl(a+^b), Q = RelIncl(X\/Y);
    set R = RelIncl ord-type(X\/Y);
A2: On(X\/Y) = X\/Y & On X = X & On Y = Y by Th2; then
A3: h is_isomorphism_of R,Q by Th21;
A4: f^g is_isomorphism_of P,Q by A1,Th24; then
A5: P,Q are_isomorphic & R,Q are_isomorphic by A3; then
    Q,R are_isomorphic by WELLORD1:40; then
    a+^b = ord-type(X\/Y) by A5,WELLORD1:42,WELLORD2:10;
    hence numbering(X \/ Y) = (numbering X)^(numbering Y)
    by A2,A4,A5,WELLORD1:def 9;
  end;
