let F1, F2 be Function of NAT , ExtREAL ; :: thesis: ( ( for n being Element of NAT holds F1 . n <= F2 . n ) implies SUM F1 <= SUM F2 )
assume A2: for n being Element of NAT holds F1 . n <= F2 . n ; :: thesis: SUM F1 <= SUM F2
for x being ext-real number st x in rng (Ser F1) holds
ex y being ext-real number st
( y in rng (Ser F2) & x <= y )

proof

let x be ext-real number ; :: thesis: ( x in rng (Ser F1) implies ex y being ext-real number st
( y in rng (Ser F2) & x <= y ) )
assume A3: x in rng (Ser F1) ; :: thesis: ex y being ext-real number st
( y in rng (Ser F2) & x <= y )
A4: ( dom (Ser F1) = NAT & dom (Ser F2) = NAT & rng (Ser F2) c= ExtREAL ) by FUNCT_2:def 1;
then consider n being set such that
A5: ( n in NAT & x = (Ser F1) . n ) by A3, FUNCT_1:def 5;
reconsider n = n as Element of NAT by A5;
reconsider y = (Ser F2) . n as R_eal ;
take y ; :: thesis: ( y in rng (Ser F2) & x <= y )
thus ( y in rng (Ser F2) & x <= y ) by A2, A4, A5, Th61, FUNCT_1:def 5; :: thesis: verum

end;

hence SUM F1 <= SUM F2 by XXREAL_2:63; :: thesis: verum