let x0 be Real; 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; ( 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) )
; ( f1 + f2 is_convergent_in x0 & lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0)) )
A4:
now 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;
( 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}
;
( (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;
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;
verum end;
hence
f1 + f2 is_convergent_in x0
by A3; lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0))
hence
lim ((f1 + f2),x0) = (lim (f1,x0)) + (lim (f2,x0))
by A4, Def4; verum