reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem Th5:
  i divides j & j divides i implies i = j
proof
  assume that
A1: i divides j and
A2: j divides i;
A3: j = i * (j div i) by A1,Th3;
  i = j * (i div j) by A2,Th3;
  then
A4: i*1 = i * (j div i) * (i div j) by A3
    .= i * ((j div i) * (i div j));
A5: i = 0 implies i = j by A3;
  (j div i) * (i div j) = 1 implies i = j
  proof
    assume (j div i) * (i div j) = 1;
    then j div i = 1 by NAT_1:15;
    hence thesis by A3;
  end;
  hence thesis by A4,A5,XCMPLX_1:5;
end;
