set IT = { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;
A1: { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } is functional

proof

let x be set ; :: according to FUNCT_1:def 19 :: thesis: ( not x in { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } or x is set )
assume x in { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ; :: thesis: x is set
then ex t being Element of Funcs NAT ,REAL st
( x = t & ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ) ;
hence x is set ; :: thesis: verum

end;

A2: f is Element of Funcs NAT ,REAL by FUNCT_2:11;
consider N being Element of NAT such that
A3: for n being Element of NAT st n >= N holds
f . n >= 0 by Def4;
for n being Element of NAT st n >= N & n in X holds
( f . n <= 1 * (f . n) & f . n >= 0 ) by A3;
then f in { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } by A2;
then reconsider IT1 = { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } as functional non empty set by A1;

now

let x be Element of IT1; :: thesis: x is Function of NAT ,REAL
x in { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;
then consider t being Element of Funcs NAT ,REAL such that
A4: x = t and
ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) ;
thus x is Function of NAT ,REAL by A4; :: thesis: verum

end;

hence { t where t is Element of Funcs NAT ,REAL : ex c being Real ex N being Element of NAT st
( c > 0 & ( for n being Element of NAT st n >= N & n in X holds
( t . n <= c * (f . n) & t . n >= 0 ) ) ) } is FUNCTION_DOMAIN of NAT , REAL by FUNCT_2:def 13; :: thesis: verum