defpred S₁[ Nat, Real] means ( x0 - (1 / ($1 + 1)) < $2 & $2 < x0 & $2 in dom f );
let g1, g2 be Real; :: thesis:
assume that
A2: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds
( f /* seq is convergent & lim (f /* seq) = g1 ) and
A3: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (left_open_halfline x0) holds
( f /* seq is convergent & lim (f /* seq) = g2 ) ; :: thesis:

let n be Element of NAT ; :: thesis:
x0 - (1 / (n + 1)) < x0 by Lm3;
then consider g being Real such that
A5: x0 - (1 / (n + 1)) < g and
A6: g < x0 and
A7: g in dom f by A1;
reconsider g = g as Element of REAL by XREAL_0:def 1;
take g = g; :: thesis:
thus S₁[n,g] by A5, A6, A7; :: thesis:

end;

consider s being Real_Sequence such that
A8: for n being Element of NAT holds S₁[n,s . n] from FUNCT_2:sch 3(A4);
A9: for n being Nat holds S₁[n,s . n]

end;