reserve i,n,m for Nat,
        r,s for Real,
        A for non empty closed_interval Subset of REAL;

theorem Th4:
  dom ( #Z n / ( #Z 0 + #Z 2)) = REAL &
      ( #Z n / ( #Z 0 + #Z 2)) is continuous &
      ( #Z n / ( #Z 0 + #Z 2)).r = (r|^n)/(1+r^2)
proof
  set Z0=#Z 0,Z2=#Z 2,Zn=#Z n, f=Zn/(Z0+Z2);
A1: dom Zn=REAL=dom (Z0+Z2) by FUNCT_2:def 1;
  hence
A2: dom f = REAL/\(REAL\{}) by Lm6,RFUNCT_1:def 1
         .= REAL;
A3:Zn|REAL is continuous & (Z0+Z2)|REAL is continuous;
  REAL c= dom Zn /\ dom (Z0+Z2) by A1;
  then f | REAL is continuous by Lm6,A3,FCONT_1:24;
  hence f is  continuous;
  Zn.r = r #Z n by TAYLOR_1:def 1
      .= (r|^n) by PREPOWER:36;
  hence f.r = (r|^n) * ((Z0+Z2).r)" by XREAL_0:def 1,A2,RFUNCT_1:def 1
           .= (r|^n)/(1+r^2) by Lm5;
end;
