 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 Th12:
  r is irrational implies
    (c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n) > 0 &
    (c_d(r).(n+1)*rfs(r).(n+2) - c_d(r).n) > 0
  proof
    assume
A1: r is irrational; then
A2: c_d(r).(n+1) >= 1 by Th8;
    rfs(r).(n+1+1) > 1 by A1,Th4; then
A4: rfs(r).(n+2)*c_d(r).(n+1) > 1*c_d(r).(n+1) by A2,XREAL_1:68; then
A5: c_d(r).(n+1)*rfs(r).(n+2) > 1 by A2,XXREAL_0:2;
A6: c_d(r).n + 1 >= 1 + 1 by A1,Th8,XREAL_1:6;
    c_d(r).(n+1) >= c_d(r).n by A1,Th7; then
    c_d(r).(n+1)*rfs(r).(n+2) > c_d(r).n by A4,XXREAL_0:2; then
    c_d(r).(n+1)*rfs(r).(n+2)-c_d(r).n > c_d(r).n-c_d(r).n by XREAL_1:14;
    hence thesis by A5,A6,XREAL_1:6;
  end;
