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

theorem
  A in B*^C implies A div^ C in B & A mod^ C in C
proof
A1: A = (A div^ C)*^C+^(A mod^ C) by Th65;
  assume
A2: A in B*^C;
  then C <> {} by ORDINAL2:38;
  then
A3: ex D st A = (A div^ C)*^C+^D & D in C by Def6;
  then
A4: (A div^ C)*^C c= A by Th24;
  assume not thesis;
  then B*^C c= (A div^ C)*^C by A3,A1,Th21,ORDINAL1:16,ORDINAL2:41;
  hence contradiction by A2,A4,ORDINAL1:5;
end;
