consider Intf being PartFunc of REAL,REAL such that

A2: ( dom Intf = right_closed_halfline a & ( for x being Real st x in dom Intf holds

Intf . x = integral (f,a,x) ) & Intf is convergent_in+infty ) by A1;

take lim_in+infty Intf ; :: thesis: ex Intf being PartFunc of REAL,REAL st

( dom Intf = right_closed_halfline a & ( for x being Real st x in dom Intf holds

Intf . x = integral (f,a,x) ) & Intf is convergent_in+infty & lim_in+infty Intf = lim_in+infty Intf )

thus ex Intf being PartFunc of REAL,REAL st

( dom Intf = right_closed_halfline a & ( for x being Real st x in dom Intf holds

Intf . x = integral (f,a,x) ) & Intf is convergent_in+infty & lim_in+infty Intf = lim_in+infty Intf ) by A2; :: thesis: verum

A2: ( dom Intf = right_closed_halfline a & ( for x being Real st x in dom Intf holds

Intf . x = integral (f,a,x) ) & Intf is convergent_in+infty ) by A1;

take lim_in+infty Intf ; :: thesis: ex Intf being PartFunc of REAL,REAL st

( dom Intf = right_closed_halfline a & ( for x being Real st x in dom Intf holds

Intf . x = integral (f,a,x) ) & Intf is convergent_in+infty & lim_in+infty Intf = lim_in+infty Intf )

thus ex Intf being PartFunc of REAL,REAL st

( dom Intf = right_closed_halfline a & ( for x being Real st x in dom Intf holds

Intf . x = integral (f,a,x) ) & Intf is convergent_in+infty & lim_in+infty Intf = lim_in+infty Intf ) by A2; :: thesis: verum