let F be Function of NAT,ExtREAL; :: thesis: ( ( for n being Element of NAT holds F . n in REAL ) implies for n being Element of NAT holds (Ser F) . n in REAL )
defpred S₁[ Element of NAT ] means (Ser F) . $1 in REAL ;
assume A1: for n being Element of NAT holds F . n in REAL ; :: thesis: for n being Element of NAT holds (Ser F) . n in REAL
A2: for s being Element of NAT st S₁[s] holds
S₁[s + 1]

proof

let s be Element of NAT ; :: thesis: ( S₁[s] implies S₁[s + 1] )
reconsider b = F . (s + 1) as Real by A1;
reconsider y = (Ser F) . s as R_eal ;
assume (Ser F) . s in REAL ; :: thesis: S₁[s + 1]
then reconsider a = y as Real ;
A3: y + (F . (s + 1)) = a + b by XXREAL_3:def 2;
(Ser F) . (s + 1) = y + (F . (s + 1)) by Th63;
hence S₁[s + 1] by A3; :: thesis: verum

end;

(Ser F) . 0 = F . 0 by Th63;
then A4: S₁[ 0 ] by A1;
thus for s being Element of NAT holds S₁[s] from NAT_1:sch 1(A4, A2); :: thesis: verum