let T, S be RealNormSpace; :: thesis: for f1, f2 being PartFunc of S,T
for x0 being Point of S st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let f1, f2 be PartFunc of S,T; :: thesis: for x0 being Point of S st f1 is_continuous_in x0 & f2 is_continuous_in x0 holds
( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
let x0 be Point of S; :: thesis: ( f1 is_continuous_in x0 & f2 is_continuous_in x0 implies ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 ) )
assume A1: ( f1 is_continuous_in x0 & f2 is_continuous_in x0 ) ; :: thesis: ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
then A2: ( x0 in dom f1 & ( for s1 being sequence of S st rng s1 c= dom f1 & s1 is convergent & lim s1 = x0 holds
( f1 /* s1 is convergent & f1 /. x0 = lim (f1 /* s1) ) ) ) by Def9;
A3: ( x0 in dom f2 & ( for s1 being sequence of S st rng s1 c= dom f2 & s1 is convergent & lim s1 = x0 holds
( f2 /* s1 is convergent & f2 /. x0 = lim (f2 /* s1) ) ) ) by A1, Def9;

now

x0 in (dom f1) /\ (dom f2) by A2, A3, XBOOLE_0:def 4;
hence A4: x0 in dom (f1 + f2) by VFUNCT_1:def 1; :: thesis: for s1 being sequence of S st rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 holds
( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
let s1 be sequence of S; :: thesis: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 implies ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) ) )
assume A5: ( rng s1 c= dom (f1 + f2) & s1 is convergent & lim s1 = x0 ) ; :: thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
then A6: rng s1 c= (dom f1) /\ (dom f2) by VFUNCT_1:def 1;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 1;
then dom (f1 + f2) c= dom f1 by XBOOLE_1:17;
then rng s1 c= dom f1 by A5, XBOOLE_1:1;
then A7: ( f1 /* s1 is convergent & f1 /. x0 = lim (f1 /* s1) ) by A1, A5, Def9;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 1;
then dom (f1 + f2) c= dom f2 by XBOOLE_1:17;
then rng s1 c= dom f2 by A5, XBOOLE_1:1;
then A8: ( f2 /* s1 is convergent & f2 /. x0 = lim (f2 /* s1) ) by A1, A5, Def9;
then (f1 /* s1) + (f2 /* s1) is convergent by A7, NORMSP_1:34;
hence (f1 + f2) /* s1 is convergent by A6, Th19; :: thesis: (f1 + f2) /. x0 = lim ((f1 + f2) /* s1)
thus (f1 + f2) /. x0 = (f1 /. x0) + (f2 /. x0) by A4, VFUNCT_1:def 1
.= lim ((f1 /* s1) + (f2 /* s1)) by A7, A8, NORMSP_1:42
.= lim ((f1 + f2) /* s1) by A6, Th19 ; :: thesis: verum

end;

hence f1 + f2 is_continuous_in x0 by Def9; :: thesis: f1 - f2 is_continuous_in x0

now

x0 in (dom f1) /\ (dom f2) by A2, A3, XBOOLE_0:def 4;
hence A9: x0 in dom (f1 - f2) by VFUNCT_1:def 2; :: thesis: for s1 being sequence of S st rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 holds
( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
let s1 be sequence of S; :: thesis: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 implies ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) ) )
assume A10: ( rng s1 c= dom (f1 - f2) & s1 is convergent & lim s1 = x0 ) ; :: thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
then A11: rng s1 c= (dom f1) /\ (dom f2) by VFUNCT_1:def 2;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 2;
then dom (f1 - f2) c= dom f1 by XBOOLE_1:17;
then rng s1 c= dom f1 by A10, XBOOLE_1:1;
then A12: ( f1 /* s1 is convergent & f1 /. x0 = lim (f1 /* s1) ) by A1, A10, Def9;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 2;
then dom (f1 - f2) c= dom f2 by XBOOLE_1:17;
then rng s1 c= dom f2 by A10, XBOOLE_1:1;
then A13: ( f2 /* s1 is convergent & f2 /. x0 = lim (f2 /* s1) ) by A1, A10, Def9;
then (f1 /* s1) - (f2 /* s1) is convergent by A12, NORMSP_1:35;
hence (f1 - f2) /* s1 is convergent by A11, Th19; :: thesis: (f1 - f2) /. x0 = lim ((f1 - f2) /* s1)
thus (f1 - f2) /. x0 = (f1 /. x0) - (f2 /. x0) by A9, VFUNCT_1:def 2
.= lim ((f1 /* s1) - (f2 /* s1)) by A12, A13, NORMSP_1:43
.= lim ((f1 - f2) /* s1) by A11, Th19 ; :: thesis: verum

end;

hence f1 - f2 is_continuous_in x0 by Def9; :: thesis: verum