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
  for X,Y being ordinal-membered set
  st for x,y st x in X & y in Y holds x in y
  holds ord-type(X \/ Y) = (ord-type X)+^(ord-type Y)
  proof
    let X,Y be ordinal-membered set;
    assume for x,y st x in X & y in Y holds x in y; then
A1: numbering(X \/ Y) = (numbering X)^(numbering Y) by Th25;
    thus ord-type(X \/ Y) = dom numbering(X \/ Y) by Th18
    .= (dom numbering X)+^(dom numbering Y) by A1,ORDINAL4:def 1
    .= (ord-type X)+^(dom numbering Y) by Th18
    .= (ord-type X)+^(ord-type Y) by Th18;
  end;
