let n be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL
for f being Function of A,(REAL n)
for g being Function of A,(REAL-NS n) st f = g & f is bounded holds
( f is integrable iff g is integrable )
let A be non empty closed_interval Subset of REAL; :: thesis: for f being Function of A,(REAL n)
for g being Function of A,(REAL-NS n) st f = g & f is bounded holds
( f is integrable iff g is integrable )
let f be Function of A,(REAL n); :: thesis: for g being Function of A,(REAL-NS n) st f = g & f is bounded holds
( f is integrable iff g is integrable )
let g be Function of A,(REAL-NS n); :: thesis: ( f = g & f is bounded implies ( f is integrable iff g is integrable ) )
assume A1: ( f = g & f is bounded ) ; :: thesis: ( f is integrable iff g is integrable )

hereby :: thesis: ( g is integrable implies f is integrable )

assume f is integrable ; :: thesis: g is integrable
then consider I being Element of REAL n such that
A2: for T being DivSequence of A
for S being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S) is convergent & lim (middle_sum (f,S)) = I ) by A1, INTEGR15:12;
reconsider I0 = I as Point of (REAL-NS n) by REAL_NS1:def 4;
I = I0 ;
then for T being DivSequence of A
for S0 being middle_volume_Sequence of g,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (g,S0) is convergent & lim (middle_sum (g,S0)) = I0 ) by A2, A1, Th40;
hence g is integrable ; :: thesis: verum

end;

assume g is integrable ; :: thesis: f is integrable
then consider I being Point of (REAL-NS n) such that
A3: for T being DivSequence of A
for S being middle_volume_Sequence of g,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (g,S) is convergent & lim (middle_sum (g,S)) = I ) ;
reconsider I0 = I as Element of REAL n by REAL_NS1:def 4;
I0 = I ;
then for T being DivSequence of A
for S0 being middle_volume_Sequence of f,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (f,S0) is convergent & lim (middle_sum (f,S0)) = I0 ) by A3, A1, Th40;
hence f is integrable by A1, INTEGR15:12; :: thesis: verum