 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 Th14:
  r is irrational implies
    r = (c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n)/
      (c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n)
  proof
    assume
A1: r is irrational;
    defpred P[Nat] means
      r = (c_n(r).($1+1)*rfs(r).($1+2)+c_n(r).$1)
    /(c_d(r).($1+1)*rfs(r).($1+2)+c_d(r).$1);
A2: P[0]
    proof
A3:   rfs(r).1 = scf(r).1 + 1/rfs(r).(1+1) by Th1;
A4:  (c_n(r).1*rfs(r).2+c_n(r).0)
      = (scf(r).1 * scf(r).0 + 1) *rfs(r).2+c_n(r).0 by REAL_3:def 5
     .= (scf(r).1 * scf(r).0 + 1) *rfs(r).2 + scf(r).0 by REAL_3:def 5
     .= scf(r).0*(scf(r).1 *rfs(r).2 + 1) + rfs(r).2;
A5:  (c_d(r).1*rfs(r).2+c_d(r).0)=scf(r).1*rfs(r).2+c_d(r).0 by REAL_3:def 6
     .= scf(r).1*rfs(r).2 + 1 by REAL_3:def 6;
A6: rfs(r).2 <> 0 & scf(r).1*rfs(r).2 + 1 <> 0 by A1,Th3;
     (c_n(r).1*rfs(r).2+c_n(r).0)/(c_d(r).1*rfs(r).2+c_d(r).0)=
     (scf(r).0*(scf(r).1*rfs(r).2+1)+rfs(r).2)/(c_d(r).1*rfs(r).2+c_d(r).0)
     by A4
     .= scf(r).0*((scf(r).1 *rfs(r).2 + 1)/( (scf(r).1*rfs(r).2 + 1)))
      + rfs(r).2 /(scf(r).1*rfs(r).2 + 1) by A5
     .= scf(r).0 *1 + rfs(r).2  /(scf(r).1*rfs(r).2 + 1) by A6,XCMPLX_1:60
     .= scf(r).0+(rfs(r).2/rfs(r).2)/((scf(r).1*rfs(r).2+1)/rfs(r).2)
         by A6,XCMPLX_1:55
     .= scf(r).0+1/((scf(r).1*rfs(r).2+1)/rfs(r).2) by A6,XCMPLX_1:60
     .= scf(r).0 + 1/(scf(r).1*(rfs(r).2/rfs(r).2)+1/rfs(r).2)
     .= scf(r).0 + 1/(scf(r).1*1+1/rfs(r).2) by A1,Th3,XCMPLX_1:60
     .= r by Th1,A3;
     hence thesis;
     end;
A7: for n be Nat st P[n] holds P[n+1]
     proof
     let n be Nat;
     assume
A8:  P[n];
A9:    rfs(r).(n+2) = scf(r).(n+2) + 1/rfs(r).(n+2 +1) by Th1
       .= scf(r).(n+2) + 1/rfs(r).(n+3);
A10:    1 = (rfs(r).(n+3))/(rfs(r).(n+3)) by A1,Th3,XCMPLX_1:60;
A11:   c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n
     = c_n(r).(n+1)*(scf(r).(n+2) + 1/rfs(r).(n+3))+c_n(r).n by A9
    .= scf(r).(n+2) * c_n(r).(n+1) + c_n(r).n + c_n(r).(n+1)* (1/rfs(r).(n+3))
    .= c_n(r).(n+2) + c_n(r).(n+1)* (1/rfs(r).(n+3)) by REAL_3:def 5;
A12:   c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n
     = c_d(r).(n+1)*(scf(r).(n+2) + 1/rfs(r).(n+3))+c_d(r).n by A9
    .= scf(r).(n+2) * c_d(r).(n+1) + c_d(r).n + c_d(r).(n+1)* (1/rfs(r).(n+3))
    .= c_d(r).(n+2) + c_d(r).(n+1)* (1/rfs(r).(n+3)) by REAL_3:def 6;
    r=(c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n)/(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n)
     by A8
   .= (c_n(r).(n+2) + c_n(r).(n+1)* (1/rfs(r).(n+3)))/
      (c_d(r).(n+2) + c_d(r).(n+1)* (1/rfs(r).(n+3))) by A11,A12
   .= ((c_n(r).(n+2) + c_n(r).(n+1)* (1/rfs(r).(n+3)))
      /(c_d(r).(n+2) + c_d(r).(n+1)* (1/rfs(r).(n+3))))
       *(rfs(r).(n+3)/rfs(r).(n+3)) by A10
   .= ((c_n(r).(n+2) + c_n(r).(n+1)* (1/rfs(r).(n+3))) * rfs(r).(n+3))
      /((c_d(r).(n+2) + c_d(r).(n+1)* (1/rfs(r).(n+3))) * rfs(r).(n+3))
      by XCMPLX_1:76
   .= ((c_n(r).(n+2)*rfs(r).(n+3)
      +c_n(r).(n+1)* ((1/rfs(r).(n+3))*rfs(r).(n+3)) ))
      /((c_d(r).(n+2)*rfs(r).(n+3)
      +c_d(r).(n+1)* ((1/rfs(r).(n+3))*rfs(r).(n+3)) ))
   .= ((c_n(r).(n+2)*rfs(r).(n+3) +c_n(r).(n+1)*1))
      /((c_d(r).(n+2)*rfs(r).(n+3)
      +c_d(r).(n+1)*((1/rfs(r).(n+3))*rfs(r).(n+3)) )) by XCMPLX_1:106,A1,Th3
   .= ((c_n(r).(n+2)*rfs(r).(n+3) +c_n(r).(n+1)*1))
      /((c_d(r).(n+2)*rfs(r).(n+3) +c_d(r).(n+1)*1)) by XCMPLX_1:106,A1,Th3
   .= (c_n(r).(n+1+1)*rfs(r).(n+1+2) +c_n(r).(n+1))
      /(c_d(r).(n+1+1)*rfs(r).(n+1+2) +c_d(r).(n+1));
     hence thesis;
  end;
  for n be Nat holds P[n] from NAT_1:sch 2(A2,A7);
  hence thesis;
end;
