reserve a, b, k, n, m for Nat,
  i for Integer,
  r for Real,
  p for Rational,
  c for Complex,
  x for object,
  f for Function;
reserve l, n1, n2 for Nat;
reserve s1, s2 for Real_Sequence;

theorem Th74:
  (for n holds scf(r).n>0) implies cocf(r).3 = scf(r).0 + 1/(scf(r
  ).1 + 1/(scf(r).2 + 1/scf(r).3))
proof
  set s=scf(r);
A1: cocf(r).3 =c_n(r).3 * ((c_d(r))").3 by SEQ_1:8
    .=c_n(r).3 * (c_d(r).3)" by VALUED_1:10
    .=c_n(r).3 *(1/c_d(r).3)
    .=c_n(r).3 /c_d(r).3;
  assume
A2: for n holds scf(r).n>0;
  then
A3: s.1>0;
A4: c_d(r).2 =s.(0+2)*c_d(r).(0+1) + c_d(r).0 by Def6
    .=s.2*s.1+c_d(r).0 by Def6
    .=s.2*s.1+1 by Def6;
A5: c_d(r).3 =s.(1+2)*c_d(r).(1+1) + c_d(r).1 by Def6
    .=s.3*(s.2*s.1+1)+s.1 by A4,Def6
    .=s.1*(s.2*s.3 +1)+s.3;
A6: c_n(r).2 =s.(0+2) * c_n(r).(0+1) + c_n(r).0 by Def5
    .=s.2*(s.1 * s.0 +1) +c_n(r).0 by Def5
    .=s.2 * s.1 * s.0 +s.2 +s.0 by Def5;
A7: c_n(r).3 =s.(1+2) * c_n(r).(1+1) + c_n(r).1 by Def5
    .=s.3*(s.2 * s.1 * s.0 +s.2 +s.0) +(s.1 * s.0 +1) by A6,Def5
    .=(s.3*s.2 * s.1 * s.0) +s.3*s.2 +s.3*s.0 +s.1 * s.0 +1;
A8: s.2>0 by A2;
A9: s.3>0 by A2;
  then s.0 + 1/(s.1 + 1/(s.2 + 1/s.3)) =s.0 + 1/(s.1 + 1/((s.2*s.3 + 1)/s.3))
  by XCMPLX_1:113
    .=s.0 + 1/(s.1 + s.3/(s.2*s.3 + 1)) by XCMPLX_1:57
    .=s.0 + 1/((s.1*(s.2*s.3 + 1) + s.3)/(s.2*s.3 + 1)) by A8,A9,XCMPLX_1:113
    .=s.0 +(s.2*s.3 + 1)/(s.1*(s.2*s.3 + 1) + s.3) by XCMPLX_1:57
    .=(s.0*(s.1*(s.2*s.3 + 1) + s.3)+(s.2*s.3 + 1))/(s.1*(s.2*s.3 + 1) + s.3
  ) by A3,A8,A9,XCMPLX_1:113;
  hence thesis by A1,A7,A5;
end;
