let X, Y be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f1, f2 being PartFunc of C,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,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,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,V; :: thesis: ( f1 is_bounded_on X & f2 is_bounded_on Y implies f1 + f2 is_bounded_on X /\ Y )
assume that
A1: f1 is_bounded_on X and
A2: f2 is_bounded_on Y ; :: thesis: f1 + f2 is_bounded_on X /\ Y
consider r1 being Real such that
A3: for c being Element of C st c in X /\ (dom f1) holds
||.(f1 /. c).|| <= r1 by A1;
consider r2 being Real such that
A4: for c being Element of C st c in Y /\ (dom f2) holds
||.(f2 /. c).|| <= r2 by A2;
take r = r1 + r2; :: according to VFUNCT_1:def 6 :: 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 )
assume A5: c in (X /\ Y) /\ (dom (f1 + f2)) ; :: thesis: ||.((f1 + f2) /. c).|| <= r
then A6: c in X /\ Y by XBOOLE_0:def 4;
then A7: c in Y by XBOOLE_0:def 4;
A8: c in dom (f1 + f2) by A5, XBOOLE_0:def 4;
then A9: c in (dom f1) /\ (dom f2) by Def1;
then c in dom f2 by XBOOLE_0:def 4;
then c in Y /\ (dom f2) by A7, XBOOLE_0:def 4;
then A10: ||.(f2 /. c).|| <= r2 by A4;
A11: c in X by A6, XBOOLE_0:def 4;
c in dom f1 by A9, XBOOLE_0:def 4;
then c in X /\ (dom f1) by A11, XBOOLE_0:def 4;
then ||.(f1 /. c).|| <= r1 by A3;
then ( ||.((f1 /. c) + (f2 /. c)).|| <= ||.(f1 /. c).|| + ||.(f2 /. c).|| & ||.(f1 /. c).|| + ||.(f2 /. c).|| <= r ) by A10, NORMSP_1:def 1, XREAL_1:7;
then ||.((f1 /. c) + (f2 /. c)).|| <= r by XXREAL_0:2;
hence ||.((f1 + f2) /. c).|| <= r by A8, Def1; :: thesis: verum