let i be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume f,D1)) . i <= (PartSums (lower_volume f,D2)) . (indx D2,D1,i)
let A be closed-interval Subset of REAL ; :: thesis: for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume f,D1)) . i <= (PartSums (lower_volume f,D2)) . (indx D2,D1,i)
let D1, D2 be Division of A; :: thesis: for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume f,D1)) . i <= (PartSums (lower_volume f,D2)) . (indx D2,D1,i)
let f be Function of A,REAL ; :: thesis: ( D1 <= D2 & i in dom D1 & f | A is bounded_below implies (PartSums (lower_volume f,D1)) . i <= (PartSums (lower_volume f,D2)) . (indx D2,D1,i) )
assume that
A1: D1 <= D2 and
A2: i in dom D1 and
A3: f | A is bounded_below ; :: thesis: (PartSums (lower_volume f,D1)) . i <= (PartSums (lower_volume f,D2)) . (indx D2,D1,i)
A4: len (lower_volume f,D2) = len D2 by Def8;
i in Seg (len D1) by A2, FINSEQ_1:def 3;
then i in Seg (len (lower_volume f,D1)) by Def8;
then i in dom (lower_volume f,D1) by FINSEQ_1:def 3;
then A5: (PartSums (lower_volume f,D1)) . i = Sum ((lower_volume f,D1) | i) by Def22;
indx D2,D1,i in dom D2 by A1, A2, Def21;
then indx D2,D1,i in Seg (len (lower_volume f,D2)) by A4, FINSEQ_1:def 3;
then A6: indx D2,D1,i in dom (lower_volume f,D2) by FINSEQ_1:def 3;
i in Seg (len D1) by A2, FINSEQ_1:def 3;
then i is non empty Element of NAT by FINSEQ_1:3;
then (PartSums (lower_volume f,D1)) . i <= Sum ((lower_volume f,D2) | (indx D2,D1,i)) by A1, A2, A3, A5, Th41;
hence (PartSums (lower_volume f,D1)) . i <= (PartSums (lower_volume f,D2)) . (indx D2,D1,i) by A6, Def22; :: thesis: verum