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 & f is_integrable_on A holds
( g is_integrable_on A & integral (f,A) = integral (g,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 & f is_integrable_on A holds
( g is_integrable_on A & integral (f,A) = integral (g,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 & f is_integrable_on A holds
( g is_integrable_on A & integral (f,A) = integral (g,A) )
let g be PartFunc of REAL,(REAL-NS n); :: thesis: ( f = g & f | A is bounded & A c= dom f & f is_integrable_on A implies ( g is_integrable_on A & integral (f,A) = integral (g,A) ) )
assume A1: ( f = g & f | A is bounded & A c= dom f & f is_integrable_on A ) ; :: thesis: ( g is_integrable_on A & integral (f,A) = integral (g,A) )
hence g is_integrable_on A by Th43; :: thesis: integral (f,A) = integral (g,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: integral (f,A) = integral h by INTEGR15:14;
A3: h is bounded by A1;
h is integrable by A1, INTEGR15:13;
then integral h = integral k by A3, Th42;
hence integral (f,A) = integral (g,A) by A2, A1, INTEGR18:9; :: thesis: verum