reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem Th45:
  B <> {} implies union(A+^B) = A+^union B
proof
  assume
A1: B <> {};
A2: now
    assume not ex C st B = succ C;
    then
A3: B is limit_ordinal by ORDINAL1:29;
    then A+^B is limit_ordinal by A1,Th29;
    then union(A+^B) = A+^B;
    hence thesis by A3;
  end;
  now
    given C such that
A4: B = succ C;
    thus union(A+^B) = union succ (A+^C) by A4,ORDINAL2:28
      .= A+^C by ORDINAL2:2
      .= A+^union B by A4,ORDINAL2:2;
  end;
  hence thesis by A2;
end;
