 reserve i,j,k,m,n,m1,n1 for Nat;
 reserve a,r,r1,r2 for Real;
 reserve m0,cn,cd for Integer;
 reserve x1,x2,o for object;

theorem Th19:
  r is irrational implies |. r - c_n(r).n/c_d(r).n .| > 0
  proof
    assume
A1: r is irrational;
    assume not |. r - c_n(r).n/c_d(r).n .| > 0; then
    |. r - c_n(r).n/c_d(r).n .| = 0 by COMPLEX1:46; then
A3: r - c_n(r).n/c_d(r).n = 0 by COMPLEX1:45;
    c_d(r).n in NAT by REAL_3:50;
    hence contradiction by A1,A3;
  end;
