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 Function of A,(REAL n) st f | A = g holds
integral (f,A) = integral g

let A be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of REAL,(REAL n)
for g being Function of A,(REAL n) st f | A = g holds
integral (f,A) = integral g

let f be PartFunc of REAL,(REAL n); :: thesis: for g being Function of A,(REAL n) st f | A = g holds
integral (f,A) = integral g

let g be Function of A,(REAL n); :: thesis: ( f | A = g implies integral (f,A) = integral g )
assume A1: f | A = g ; :: thesis: integral (f,A) = integral g
A2: now :: thesis: for k being Nat st k in dom (integral (f,A)) holds
(integral (f,A)) . k = () . k
let k be Nat; :: thesis: ( k in dom (integral (f,A)) implies (integral (f,A)) . k = () . k )
assume A3: k in dom (integral (f,A)) ; :: thesis: (integral (f,A)) . k = () . k
then reconsider i = k as Element of NAT ;
dom (proj (i,n)) = REAL n by FUNCT_2:def 1;
then rng f c= dom (proj (i,n)) ;
then A4: dom ((proj (i,n)) * f) = dom f by RELAT_1:27;
A = dom g by FUNCT_2:def 1
.= (dom f) /\ A by ;
then ((proj (i,n)) * f) || A is total by ;
then reconsider F = ((proj (i,n)) * f) | A as Function of A,REAL ;
A5: F = (proj (i,n)) * g by ;
A6: i in Seg n by ;
then (integral (f,A)) . i = integral (((proj (i,n)) * f),A) by Def17
.= integral (((proj (i,n)) * f) || A) ;
hence (integral (f,A)) . k = () . k by ; :: thesis: verum
end;
dom (integral (f,A)) = Seg n by Def17
.= dom () by Def14 ;
hence integral (f,A) = integral g by ; :: thesis: verum