reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;

theorem Th7:
  for n,m holds m divides n iff n = m *^ (n div^ m)
proof
  let n,m;
  assume
A1: not thesis;
  then consider a such that
A2: n = m*^a by Th5;
  {}*^a = {} by ORDINAL2:35;
  then {} <> m by A1,A2,ORDINAL2:35;
  then
A3: {} in m by ORDINAL3:8;
  n = a*^m+^{} by A2,ORDINAL2:27;
  hence thesis by A1,A2,A3,ORDINAL3:def 6;
end;
