let i be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
let A be non empty closed_interval Subset of REAL; :: thesis: for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
let D be Division of A; :: thesis: for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
let f, g be Function of A,REAL; :: thesis: ( i in dom D & f | A is bounded_below & g | A is bounded_below implies ((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i )
assume that
A1: i in dom D and
A2: f | A is bounded_below and
A3: g | A is bounded_below ; :: thesis: ((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
A4: 0 <= vol (divset (D,i)) by SEQ_4:11, XREAL_1:48;
dom (f + g) = A /\ A by FUNCT_2:def 1;
then dom ((f + g) | (divset (D,i))) = divset (D,i) by A1, Th6, RELAT_1:62;
then A5: not rng ((f + g) | (divset (D,i))) is empty by RELAT_1:42;
rng g is bounded_below by A3, Th9;
then A6: rng (g | (divset (D,i))) is bounded_below by RELAT_1:70, XXREAL_2:44;
dom g = A by FUNCT_2:def 1;
then dom (g | (divset (D,i))) = divset (D,i) by A1, Th6, RELAT_1:62;
then A7: not rng (g | (divset (D,i))) is empty by RELAT_1:42;
(f + g) | (divset (D,i)) = (f | (divset (D,i))) + (g | (divset (D,i))) by RFUNCT_1:44;
then A8: rng ((f + g) | (divset (D,i))) c= (rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i)))) by Th8;
rng f is bounded_below by A2, Th9;
then A9: rng (f | (divset (D,i))) is bounded_below by RELAT_1:70, XXREAL_2:44;
then A10: (rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i)))) is bounded_below by A6, SEQ_4:124;
dom f = A by FUNCT_2:def 1;
then dom (f | (divset (D,i))) = divset (D,i) by A1, Th6, RELAT_1:62;
then not rng (f | (divset (D,i))) is empty by RELAT_1:42;
then lower_bound ((rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i))))) = (lower_bound (rng (f | (divset (D,i))))) + (lower_bound (rng (g | (divset (D,i))))) by A9, A6, A7, SEQ_4:125;
then (lower_bound (rng ((f + g) | (divset (D,i))))) * (vol (divset (D,i))) >= ((lower_bound (rng (f | (divset (D,i))))) + (lower_bound (rng (g | (divset (D,i)))))) * (vol (divset (D,i))) by A10, A5, A4, A8, SEQ_4:47, XREAL_1:64;
then (lower_volume ((f + g),D)) . i >= ((lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))) + ((lower_bound (rng (g | (divset (D,i))))) * (vol (divset (D,i)))) by A1, Def6;
then (lower_volume ((f + g),D)) . i >= ((lower_volume (f,D)) . i) + ((lower_bound (rng (g | (divset (D,i))))) * (vol (divset (D,i)))) by A1, Def6;
hence ((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i by A1, Def6; :: thesis: verum