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

theorem Th8:
  for n,m st n divides m & m divides n holds n = m
proof
  let n,m;
  assume that
A1: n divides m and
A2: m divides n;
A3: m = n *^ (m div^ n) by A1,Th7;
A4: (m div^ n) *^ (n div^ m) = 1 implies n = m
  proof
    assume (m div^ n) *^ (n div^ m) = 1;
    then m div^ n = 1 by ORDINAL3:37;
    hence thesis by A3,ORDINAL2:39;
  end;
  n = m *^ (n div^ m) by A2,Th7;
  then
A5: n *^ 1 = n & n = n *^ ((m div^ n) *^ (n div^ m)) by A3,ORDINAL2:39
,ORDINAL3:50;
  n = {} implies n = m by A3,ORDINAL2:35;
  hence thesis by A5,A4,ORDINAL3:33;
end;
