let A be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of REAL,COMPLEX

for g being Function of A,COMPLEX st f | A = g holds

integral (f,A) = integral g

let f be PartFunc of REAL,COMPLEX; :: thesis: for g being Function of A,COMPLEX st f | A = g holds

integral (f,A) = integral g

let g be Function of A,COMPLEX; :: thesis: ( f | A = g implies integral (f,A) = integral g )

assume A1: f | A = g ; :: thesis: integral (f,A) = integral g

A2: A = dom g by FUNCT_2:def 1

.= (dom f) /\ A by A1, RELAT_1:61 ;

then A = (dom (Re f)) /\ A by COMSEQ_3:def 3;

then (Re f) || A is total by INTEGRA5:6, XBOOLE_1:17;

then reconsider F = (Re f) | A as Function of A,REAL ;

dom g = A by FUNCT_2:def 1;

then reconsider g0 = g as PartFunc of REAL,COMPLEX by RELSET_1:5;

A3: Re g = Re g0

.= F by A1, Lm4 ;

A = (dom (Im f)) /\ A by A2, COMSEQ_3:def 4;

then (Im f) || A is total by INTEGRA5:6, XBOOLE_1:17;

then reconsider G = (Im f) | A as Function of A,REAL ;

Im g = Im g0

.= G by A1, Lm4 ;

hence integral (f,A) = integral g by A3; :: thesis: verum

for g being Function of A,COMPLEX st f | A = g holds

integral (f,A) = integral g

let f be PartFunc of REAL,COMPLEX; :: thesis: for g being Function of A,COMPLEX st f | A = g holds

integral (f,A) = integral g

let g be Function of A,COMPLEX; :: thesis: ( f | A = g implies integral (f,A) = integral g )

assume A1: f | A = g ; :: thesis: integral (f,A) = integral g

A2: A = dom g by FUNCT_2:def 1

.= (dom f) /\ A by A1, RELAT_1:61 ;

then A = (dom (Re f)) /\ A by COMSEQ_3:def 3;

then (Re f) || A is total by INTEGRA5:6, XBOOLE_1:17;

then reconsider F = (Re f) | A as Function of A,REAL ;

dom g = A by FUNCT_2:def 1;

then reconsider g0 = g as PartFunc of REAL,COMPLEX by RELSET_1:5;

A3: Re g = Re g0

.= F by A1, Lm4 ;

A = (dom (Im f)) /\ A by A2, COMSEQ_3:def 4;

then (Im f) || A is total by INTEGRA5:6, XBOOLE_1:17;

then reconsider G = (Im f) | A as Function of A,REAL ;

Im g = Im g0

.= G by A1, Lm4 ;

hence integral (f,A) = integral g by A3; :: thesis: verum