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

proof

let x be object ; :: according to FUNCT_1:def 13 :: thesis: ( not x in { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } or x is set )
assume x in { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } ; :: thesis: x is set
then ex t being Element of Funcs (NAT,REAL) st
( x = t & ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) ) ;
hence x is set ; :: thesis: verum

end;

( f is Element of Funcs (NAT,REAL) & ( for n being Element of NAT st n >= 0 & n in X holds
( 1 * (f . n) <= f . n & f . n <= 1 * (f . n) ) ) ) by FUNCT_2:8;
then f in { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } ;
then reconsider IT1 = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } as non empty functional set by A1;

now :: thesis: for x being Element of IT1 holds x is sequence of REAL

let x be Element of IT1; :: thesis: x is sequence of REAL
x in { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } ;
then ex t being Element of Funcs (NAT,REAL) st
( x = t & ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) ) ;
hence x is sequence of REAL ; :: thesis: verum

end;

hence { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st
( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds
( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } is FUNCTION_DOMAIN of NAT , REAL by FUNCT_2:def 12; :: thesis: verum