let n be Element of NAT ; :: thesis: for a, b being Real
for f being PartFunc of REAL,(REAL n)
for g being PartFunc of REAL,() st f = g & a <= b & f | ['a,b'] is bounded & ['a,b'] c= dom f & f is_integrable_on ['a,b'] holds
integral (f,a,b) = integral (g,a,b)

let a, b be Real; :: thesis: for f being PartFunc of REAL,(REAL n)
for g being PartFunc of REAL,() st f = g & a <= b & f | ['a,b'] is bounded & ['a,b'] c= dom f & f is_integrable_on ['a,b'] holds
integral (f,a,b) = integral (g,a,b)

let f be PartFunc of REAL,(REAL n); :: thesis: for g being PartFunc of REAL,() st f = g & a <= b & f | ['a,b'] is bounded & ['a,b'] c= dom f & f is_integrable_on ['a,b'] holds
integral (f,a,b) = integral (g,a,b)

let g be PartFunc of REAL,(); :: thesis: ( f = g & a <= b & f | ['a,b'] is bounded & ['a,b'] c= dom f & f is_integrable_on ['a,b'] implies integral (f,a,b) = integral (g,a,b) )
assume A1: ( f = g & a <= b & f | ['a,b'] is bounded & ['a,b'] c= dom f & f is_integrable_on ['a,b'] ) ; :: thesis: integral (f,a,b) = integral (g,a,b)
A2: ['a,b'] = [.a,b.] by ;
A3: integral (g,['a,b']) = integral (f,['a,b']) by ;
integral (g,['a,b']) = integral (g,a,b) by ;
hence integral (f,a,b) = integral (g,a,b) by ; :: thesis: verum