 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
  r is irrational implies cocf(r) is convergent & lim (cocf(r)) = r
  proof
    assume
A1: r is irrational;
A2: for p be Real st 0<p ex n st for m st n<=m holds |. cocf(r).m-r.| < p
    proof
      let p be Real;
      assume
A3:   p>0; then
A4:   sqrt p > 0 by SQUARE_1:25;
A7:   [/ 1/ sqrt p \] >= 1/sqrt p by INT_1:def 7;
      [/ 1/ sqrt p \] > 0 by A4,INT_1:def 7; then
      [/ 1/ sqrt p \] in NAT by INT_1:3; then
      consider n such that
A9:   n = [/ 1/ sqrt p \];
A10:  n > 0 by A4,INT_1:def 7,A9;
      for k be Nat st k >= n holds |. cocf(r).k-r.| < p
      proof
        let k be Nat;
        assume
A12:    k >= n;
A13:    cocf(r).k =( c_n(r) /" c_d(r)).k by REAL_3:def 7
         .= c_n(r).k * ((c_d(r))").k by SEQ_1:8
         .= c_n(r).k /c_d(r).k by VALUED_1:10;
A14:    |.cocf(r).k - r.| = |.-(cocf(r).k - r).| by COMPLEX1:52
         .= |.r - c_n(r).k /c_d(r).k.| by A13;
        (1/sqrt p)^2 = (sqrt p)"*(sqrt p)" by SQUARE_1:def 1
         .= (sqrt p*sqrt p)" by XCMPLX_1:204 .= (sqrt p)^2" by SQUARE_1:def 1
         .= p" by A3,SQUARE_1:def 2; then
        n^2 >= p" by A4,A7,SQUARE_1:15,A9; then
A16:    (n^2)" <= (p")" by A3,XREAL_1:85;
        k^2 >= n^2 by A12,SQUARE_1:15; then
A18:    (k^2)" <= (n^2)" by A10,SQUARE_1:12,XREAL_1:85;
A19:    (c_d(r).k)^2 =(c_d(r).k)*(c_d(r).k) by SQUARE_1:def 1
        .= c_d(r).k|^2 by WSIERP_1:1;
A20:    (c_d(r).k)^2 >= k^2 by A1,Th10,SQUARE_1:15;
        ((c_d(r).k)^2)"=(c_d(r).k|^2)" by A19 .= 1/c_d(r).k|^2; then
        1/c_d(r).k|^2 <= (k^2)"
          by A12,A10,SQUARE_1:12,XREAL_1:85,A20; then
        1/c_d(r).k|^2 <= (n^2)" by A18,XXREAL_0:2;
        then 1/c_d(r).k|^2 <= p by A16,XXREAL_0:2;
        hence thesis by XXREAL_0:2,A1,Th24,A14;
      end;
      hence thesis;
    end;
    cocf(r) is convergent by A2,SEQ_2:def 6;
    hence thesis by A2,SEQ_2:def 7;
  end;
