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 13 :: 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;

consider N being Element of NAT such that
A2: for n being Element of NAT st n >= N holds
f . n >= 0 by Def4;
( f is Element of Funcs (NAT,REAL) & ( for n being Element of NAT st n >= N & n in X holds
( f . n <= 1 * (f . n) & f . n >= 0 ) ) ) by A2, FUNCT_2:8;
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 ) ) ) } ;
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 non empty functional 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 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 Function of NAT,REAL ; :: 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 12; :: thesis: verum