 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 Th4:
  r is irrational implies
    scf(r).n < rfs(r).n & rfs(r).n < scf(r).n + 1 & 1 < rfs(r).(n+1)
  proof
    assume r is irrational; then
A2: rfs(r).n is not integer by Th2;
A3: 1/rfs(r).(n+1) = 1/(1/frac(rfs(r).n)) by REAL_3:def 3
      .= frac(rfs(r).n);
A4: 1/rfs(r).(n +1)> 0 by A2,A3,INT_1:46;
A5: rfs(r).n = scf(r).n + 1/rfs(r).(n+1) by Th1;
A6: rfs(r).n - scf(r).n + scf(r).n > 0 + scf(r).n
      by A2,A3,A5,INT_1:46,XREAL_1:8;
    1" < ((rfs(r).(n +1))")" by A3,INT_1:43,A4,XREAL_1:88;
    hence thesis by A6,A5,A3,INT_1:43,XREAL_1:8;
  end;
