let X, Y be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,the carrier of V st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 + f2 is_bounded_on X /\ Y
let C be non empty set ; :: thesis: for V being RealNormSpace
for f1, f2 being PartFunc of C,the carrier of V st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 + f2 is_bounded_on X /\ Y
let V be RealNormSpace; :: thesis: for f1, f2 being PartFunc of C,the carrier of V st f1 is_bounded_on X & f2 is_bounded_on Y holds
f1 + f2 is_bounded_on X /\ Y
let f1, f2 be PartFunc of C,the carrier of V; :: thesis: ( f1 is_bounded_on X & f2 is_bounded_on Y implies f1 + f2 is_bounded_on X /\ Y )
assume A1: ( f1 is_bounded_on X & f2 is_bounded_on Y ) ; :: thesis: f1 + f2 is_bounded_on X /\ Y
then consider r1 being Real such that
A2: for c being Element of C st c in X /\ (dom f1) holds
||.(f1 /. c).|| <= r1 by Def7;
consider r2 being Real such that
A3: for c being Element of C st c in Y /\ (dom f2) holds
||.(f2 /. c).|| <= r2 by A1, Def7;
take r = r1 + r2; :: according to VFUNCT_1:def 7 :: thesis: for c being Element of C st c in (X /\ Y) /\ (dom (f1 + f2)) holds
||.((f1 + f2) /. c).|| <= r
let c be Element of C; :: thesis: ( c in (X /\ Y) /\ (dom (f1 + f2)) implies ||.((f1 + f2) /. c).|| <= r )
A4: ||.((f1 /. c) + (f2 /. c)).|| <= ||.(f1 /. c).|| + ||.(f2 /. c).|| by NORMSP_1:def 2;
assume c in (X /\ Y) /\ (dom (f1 + f2)) ; :: thesis: ||.((f1 + f2) /. c).|| <= r
then A5: ( c in X /\ Y & c in dom (f1 + f2) ) by XBOOLE_0:def 4;
then A6: ( c in X & c in Y & c in (dom f1) /\ (dom f2) ) by Def1, XBOOLE_0:def 4;
then ( c in dom f1 & c in dom f2 ) by XBOOLE_0:def 4;
then A7: ( c in X /\ (dom f1) & c in Y /\ (dom f2) ) by A6, XBOOLE_0:def 4;
then A8: ||.(f1 /. c).|| <= r1 by A2;
||.(f2 /. c).|| <= r2 by A3, A7;
then ||.(f1 /. c).|| + ||.(f2 /. c).|| <= r by A8, XREAL_1:9;
then ||.((f1 /. c) + (f2 /. c)).|| <= r by A4, XXREAL_0:2;
hence ||.((f1 + f2) /. c).|| <= r by A5, Def1; :: thesis: verum