reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;

theorem Th4:
   cn = c_n(r).n & cd = c_d(r).n implies cn,cd are_coprime
   proof
     set cd2 = c_d(r).(n+1);
     set cn2 = c_n(r).(n+1);
     assume that
A1:  cn = c_n(r).n and
A2:  cd = c_d(r).n;
A6:  cn2*cd -cn*cd2 = (-1)|^n by A1,A2,REAL_3:64;
     assume
A5:  not cn,cd are_coprime;
     consider cn0,cd0 be Integer such that
A7:  cn = (cn gcd cd)*cn0 and
A8:  cd = (cn gcd cd)*cd0 and
     cn0,cd0 are_coprime by A2,INT_2:23;
     cn gcd cd >= 0 + 1 by A2,INT_1:7; then
A9:  cn gcd cd > 1 by A5,INT_2:def 3,XXREAL_0:1;
A10: (-1)|^n = (cn gcd cd)*(cn2*cd0 - cn0*cd2) by A7,A6,A8;
A11: 1=|.(-1)|^n.| by SERIES_2:1
     .=|.(cn gcd cd).|*|.(cn2*cd0-cn0*cd2).| by COMPLEX1:65,A10
     .=(cn gcd cd)*|.(cn2*cd0-cn0*cd2).|;
     (cn gcd cd)" < 1 by A9,XREAL_1:212; then
     |.(cn2*cd0-cn0*cd2).| < 1 by A11,XCMPLX_1:210; then
     |.(cn2*cd0-cn0*cd2).| = 0 by NAT_1:14;
     hence contradiction by A11;
   end;
