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,() 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,() 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,() 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,(); :: 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 ;
reconsider k = h as Function of A,() by REAL_NS1:def 4;
A2: h is bounded by A1;
hereby :: thesis: ( g is_integrable_on A implies f is_integrable_on A ) end;
assume g is_integrable_on A ; :: thesis:
then k is integrable by A1;
then h is integrable by ;
hence f is_integrable_on A by INTEGR15:13; :: thesis: verum