let x0 be Real; :: thesis: for f1, f2 being PartFunc of REAL,REAL st f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) holds
( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( f1 is_convergent_in x0 & f2 is_convergent_in x0 & ( for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ) implies ( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) ) )
assume that
A1: f1 is_convergent_in x0 and
A2: f2 is_convergent_in x0 and
A3: for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom (f1 + f2) & g2 < r2 & x0 < g2 & g2 in dom (f1 + f2) ) ; :: thesis: ( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) )

A4: now :: thesis: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) \ {x0} holds
( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) )

let seq be Real_Sequence; :: thesis: ( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) \ {x0} implies ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) ) )
assume that
A5: seq is convergent and
A6: lim seq = x0 and
A7: rng seq c= (dom (f1 + f2)) \ {x0} ; :: thesis: ( (f1 + f2) /* seq is convergent & lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) )
A8: dom (f1 + f2) = (dom f1) /\ (dom f2) by A7, Lm4;
A9: rng seq c= (dom f1) \ {x0} by A7, Lm4;
A10: rng seq c= (dom f2) \ {x0} by A7, Lm4;
then A11: lim (f2 /* seq) = lim (f2,x0) by A2, A5, A6, Def4;
A12: f2 /* seq is convergent by A2, A5, A6, A10;
rng seq c= dom (f1 + f2) by A7, Lm4;
then A13: (f1 /* seq) + (f2 /* seq) = (f1 + f2) /* seq by A8, RFUNCT_2:8;
A14: f1 /* seq is convergent by A1, A5, A6, A9;
hence (f1 + f2) /* seq is convergent by A12, A13, SEQ_2:5; :: thesis: lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0))
lim (f1 /* seq) = lim (f1,x0) by A1, A5, A6, A9, Def4;
hence lim ((f1 + f2) /* seq) = (lim (f1,x0)) + (lim (f2,x0)) by A14, A12, A11, A13, SEQ_2:6; :: thesis: verum

end;

hence f1 + f2 is_convergent_in x0 by A3; :: thesis: lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0))
hence lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) by A4, Def4; :: thesis: verum