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

for f being PartFunc of REAL,(REAL n)

for g being PartFunc of REAL,(REAL-NS n) st f = g & f | A is bounded & A c= dom f holds

( f is_integrable_on A iff g is_integrable_on A )

let A be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of REAL,(REAL n)

for g being PartFunc of REAL,(REAL-NS n) st f = g & f | A is bounded & A c= dom f holds

( f is_integrable_on A iff g is_integrable_on A )

let f be PartFunc of REAL,(REAL n); :: thesis: for g being PartFunc of REAL,(REAL-NS n) st f = g & f | A is bounded & A c= dom f holds

( f is_integrable_on A iff g is_integrable_on A )

let g be PartFunc of REAL,(REAL-NS n); :: thesis: ( f = g & f | A is bounded & A c= dom f implies ( f is_integrable_on A iff g is_integrable_on A ) )

assume A1: ( f = g & f | A is bounded & A c= dom f ) ; :: thesis: ( f is_integrable_on A iff g is_integrable_on A )

reconsider h = f | A as Function of A,(REAL n) by Lm3, A1;

reconsider k = h as Function of A,(REAL-NS n) by REAL_NS1:def 4;

A2: h is bounded by A1;

then k is integrable by A1;

then h is integrable by A2, Th41;

hence f is_integrable_on A by INTEGR15:13; :: thesis: verum

for f being PartFunc of REAL,(REAL n)

for g being PartFunc of REAL,(REAL-NS n) st f = g & f | A is bounded & A c= dom f holds

( f is_integrable_on A iff g is_integrable_on A )

let A be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of REAL,(REAL n)

for g being PartFunc of REAL,(REAL-NS n) st f = g & f | A is bounded & A c= dom f holds

( f is_integrable_on A iff g is_integrable_on A )

let f be PartFunc of REAL,(REAL n); :: thesis: for g being PartFunc of REAL,(REAL-NS n) st f = g & f | A is bounded & A c= dom f holds

( f is_integrable_on A iff g is_integrable_on A )

let g be PartFunc of REAL,(REAL-NS n); :: thesis: ( f = g & f | A is bounded & A c= dom f implies ( f is_integrable_on A iff g is_integrable_on A ) )

assume A1: ( f = g & f | A is bounded & A c= dom f ) ; :: thesis: ( f is_integrable_on A iff g is_integrable_on A )

reconsider h = f | A as Function of A,(REAL n) by Lm3, A1;

reconsider k = h as Function of A,(REAL-NS n) by REAL_NS1:def 4;

A2: h is bounded by A1;

hereby :: thesis: ( g is_integrable_on A implies f is_integrable_on A )

assume
g is_integrable_on A
; :: thesis: f is_integrable_on Aassume
f is_integrable_on A
; :: thesis: g is_integrable_on A

then h is integrable by INTEGR15:13;

then k is integrable by A2, Th41;

hence g is_integrable_on A by A1; :: thesis: verum

end;then h is integrable by INTEGR15:13;

then k is integrable by A2, Th41;

hence g is_integrable_on A by A1; :: thesis: verum

then k is integrable by A1;

then h is integrable by A2, Th41;

hence f is_integrable_on A by INTEGR15:13; :: thesis: verum