 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
  cn = c_n(r).n & cd = c_d(r).n & cn <> 0 implies
    cn,cd are_coprime
  proof
    c_d(r).(n+1) in NAT by REAL_3:50; then
    reconsider cd2 = c_d(r).(n+1) as Integer;
    reconsider cn2 = c_n(r).(n+1) as Integer;
    assume that
A2: cn = c_n(r).n and
A3: cd = c_d(r).n and
A4: cn <> 0;
    assume
A5: not (cn,cd are_coprime);
A6: cn2*cd -cn*cd2 = (-1)|^n by A2,A3,REAL_3:64;
    consider cn0,cd0 be Integer such that
A8: cn = (cn gcd cd)*cn0 and
A9: cd = (cn gcd cd)*cd0 and
    cn0,cd0 are_coprime by A4,INT_2:23;
    cn gcd cd <> 0 by A4,INT_2:5; then
    cn gcd cd >= 0 +1 by INT_1:7; then
A11:cn gcd cd > 1 by A5,INT_2:def 3,XXREAL_0:1;
A12:(-1)|^n = cn2*(cn gcd cd)*cd0 -cn*cd2 by A6,A9
         .= (cn gcd cd)*(cn2*cd0 - cn0*cd2) by A8;
A13:1=|.(-1)|^n.| by SERIES_2:1 .=|.(cn gcd cd)*(cn2*cd0-cn0*cd2).| by A12
      .=|.(cn gcd cd).|*|.(cn2*cd0-cn0*cd2).| by COMPLEX1:65
      .=(cn gcd cd)*|.(cn2*cd0-cn0*cd2).| by ABSVALUE:def 1;
    (cn gcd cd)" < 1 by A11,XREAL_1:212; then
    |.(cn2*cd0-cn0*cd2).| < 1 by A13,XCMPLX_1:210; then
    |.(cn2*cd0-cn0*cd2).| = 0 by NAT_1:14;
    hence contradiction by A13;
  end;
