 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 Th1:
  r = rfs(r).0 & r = scf(r).0 + 1/rfs(r).1 &
    rfs(r).n = scf(r).n + 1/rfs(r).(n+1)
  proof
A1:  rfs(r).1 = rfs(r).(0+1) .= 1/frac(rfs(r).0) by REAL_3:def 3;
A2:  frac(rfs(r).0) = frac(r) by REAL_3:def 3 .= r - scf(r).0 by REAL_3:35;
A4:  frac(rfs(r).n) = rfs(r).n - [\ rfs(r).n /] by INT_1:def 8;
     1/rfs(r).(n+1) = 1/(1/frac(rfs(r).n)) by REAL_3:def 3
     .= frac(rfs(r).n); then
     scf(r).n + 1/rfs(r).(n+1) = scf(r).n + rfs(r).n - [\ rfs(r).n /] by A4
     .= scf(r).n + rfs(r).n - scf(r).n by REAL_3:def 4 .= rfs(r).n;
     hence thesis by REAL_3:def 3,A2,A1;
  end;
