theorem
  (A c= B or A in B) & C in D implies A+^C in B+^D
proof
  assume that
A1: A c= B or A in B and
A2: C in D;
A3: B+^C in B+^D by A2,ORDINAL2:32;
  A c= B by A1,ORDINAL1:def 2;
  hence thesis by A3,ORDINAL1:12,ORDINAL2:34;
end;
