let i, j be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL

for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds

indx (D1,D,i) < indx (D1,D,j)

let A be non empty closed_interval Subset of REAL; :: thesis: for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds

indx (D1,D,i) < indx (D1,D,j)

let D, D1 be Division of A; :: thesis: ( D <= D1 & i in dom D & j in dom D & i < j implies indx (D1,D,i) < indx (D1,D,j) )

assume that

A1: D <= D1 and

A2: i in dom D and

A3: j in dom D and

A4: i < j ; :: thesis: indx (D1,D,i) < indx (D1,D,j)

A5: D . i = D1 . (indx (D1,D,i)) by A1, A2, INTEGRA1:def 19;

A6: indx (D1,D,j) in dom D1 by A1, A3, INTEGRA1:def 19;

A7: D . j = D1 . (indx (D1,D,j)) by A1, A3, INTEGRA1:def 19;

A8: indx (D1,D,i) in dom D1 by A1, A2, INTEGRA1:def 19;

D . i < D . j by A2, A3, A4, SEQM_3:def 1;

hence indx (D1,D,i) < indx (D1,D,j) by A5, A8, A7, A6, SEQ_4:137; :: thesis: verum

for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds

indx (D1,D,i) < indx (D1,D,j)

let A be non empty closed_interval Subset of REAL; :: thesis: for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds

indx (D1,D,i) < indx (D1,D,j)

let D, D1 be Division of A; :: thesis: ( D <= D1 & i in dom D & j in dom D & i < j implies indx (D1,D,i) < indx (D1,D,j) )

assume that

A1: D <= D1 and

A2: i in dom D and

A3: j in dom D and

A4: i < j ; :: thesis: indx (D1,D,i) < indx (D1,D,j)

A5: D . i = D1 . (indx (D1,D,i)) by A1, A2, INTEGRA1:def 19;

A6: indx (D1,D,j) in dom D1 by A1, A3, INTEGRA1:def 19;

A7: D . j = D1 . (indx (D1,D,j)) by A1, A3, INTEGRA1:def 19;

A8: indx (D1,D,i) in dom D1 by A1, A2, INTEGRA1:def 19;

D . i < D . j by A2, A3, A4, SEQM_3:def 1;

hence indx (D1,D,i) < indx (D1,D,j) by A5, A8, A7, A6, SEQ_4:137; :: thesis: verum