 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 Th3:
  r is irrational implies rfs(r).n <> 0 &
  rfs(r).1*rfs(r).2 <> 0 &
  scf(r).1*rfs(r).2 + 1 <> 0
  proof
    assume
A1: r is irrational;
A2: rfs(r).n <> 0
    proof
      assume rfs(r).n = 0; then
      consider n1 be Nat such that
A4:   rfs(r).n1 = 0;
      n1 <= m implies scf(r).m = 0 by A4,REAL_3:28;
      hence contradiction by A1,REAL_3:42;
    end;
A5: rfs(r).1 <> 0 & rfs(r).2 <> 0 by A1,Th2;
    rfs(r).1 = scf(r).1 + 1/rfs(r).(1+1) by Th1; then
    rfs(r).1*rfs(r).2 = scf(r).1*rfs(r).2 + (1/rfs(r).2)*rfs(r).2
       .= scf(r).1*rfs(r).2 + 1 by XCMPLX_1:106,A5;
    hence thesis by A2,A5;
end;
