let S, T 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 that
A1: f1 is_continuous_in x0 and
A2: f2 is_continuous_in x0 ; :: thesis: ( f1 + f2 is_continuous_in x0 & f1 - f2 is_continuous_in x0 )
A3: ( x0 in dom f1 & x0 in dom f2 ) by A1, A2;

now :: thesis: ( x0 in dom (f1 + f2) & ( 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) ) ) )

x0 in (dom f1) /\ (dom f2) by 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 that
A5: rng s1 c= dom (f1 + f2) and
A6: ( s1 is convergent & lim s1 = x0 ) ; :: thesis: ( (f1 + f2) /* s1 is convergent & (f1 + f2) /. x0 = lim ((f1 + f2) /* s1) )
A7: rng s1 c= (dom f1) /\ (dom f2) by A5, VFUNCT_1:def 1;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 1;
then dom (f1 + f2) c= dom f2 by XBOOLE_1:17;
then A8: rng s1 c= dom f2 by A5;
then A9: f2 /* s1 is convergent by A2, A6;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 1;
then dom (f1 + f2) c= dom f1 by XBOOLE_1:17;
then A10: rng s1 c= dom f1 by A5;
then A11: f1 /* s1 is convergent by A1, A6;
then (f1 /* s1) + (f2 /* s1) is convergent by A9, NORMSP_1:19;
hence (f1 + f2) /* s1 is convergent by A7, Th12; :: thesis: (f1 + f2) /. x0 = lim ((f1 + f2) /* s1)
A12: f1 /. x0 = lim (f1 /* s1) by A1, A6, A10;
A13: f2 /. x0 = lim (f2 /* s1) by A2, A6, A8;
thus (f1 + f2) /. x0 = (f1 /. x0) + (f2 /. x0) by A4, VFUNCT_1:def 1
.= lim ((f1 /* s1) + (f2 /* s1)) by A11, A12, A9, A13, NORMSP_1:25
.= lim ((f1 + f2) /* s1) by A7, Th12 ; :: thesis: verum

end;

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

now :: thesis: ( x0 in dom (f1 - f2) & ( 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) ) ) )

x0 in (dom f1) /\ (dom f2) by A3, XBOOLE_0:def 4;
hence A14: 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 that
A15: rng s1 c= dom (f1 - f2) and
A16: ( s1 is convergent & lim s1 = x0 ) ; :: thesis: ( (f1 - f2) /* s1 is convergent & (f1 - f2) /. x0 = lim ((f1 - f2) /* s1) )
A17: rng s1 c= (dom f1) /\ (dom f2) by A15, VFUNCT_1:def 2;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 2;
then dom (f1 - f2) c= dom f2 by XBOOLE_1:17;
then A18: rng s1 c= dom f2 by A15;
then A19: f2 /* s1 is convergent by A2, A16;
dom (f1 - f2) = (dom f1) /\ (dom f2) by VFUNCT_1:def 2;
then dom (f1 - f2) c= dom f1 by XBOOLE_1:17;
then A20: rng s1 c= dom f1 by A15;
then A21: f1 /* s1 is convergent by A1, A16;
then (f1 /* s1) - (f2 /* s1) is convergent by A19, NORMSP_1:20;
hence (f1 - f2) /* s1 is convergent by A17, Th12; :: thesis: (f1 - f2) /. x0 = lim ((f1 - f2) /* s1)
A22: f1 /. x0 = lim (f1 /* s1) by A1, A16, A20;
A23: f2 /. x0 = lim (f2 /* s1) by A2, A16, A18;
thus (f1 - f2) /. x0 = (f1 /. x0) - (f2 /. x0) by A14, VFUNCT_1:def 2
.= lim ((f1 /* s1) - (f2 /* s1)) by A21, A22, A19, A23, NORMSP_1:26
.= lim ((f1 - f2) /* s1) by A17, Th12 ; :: thesis: verum

end;

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