let X be set ; :: thesis: ( X is natural-functions-membered implies X is integer-functions-membered )

assume A1: for x being object st x in X holds

x is natural-valued Function ; :: according to VALUED_2:def 7 :: thesis: X is integer-functions-membered

let x be object ; :: according to VALUED_2:def 6 :: thesis: ( x in X implies x is INT -valued Function )

assume x in X ; :: thesis: x is INT -valued Function

then x is natural-valued Function by A1;

hence x is INT -valued Function ; :: thesis: verum

assume A1: for x being object st x in X holds

x is natural-valued Function ; :: according to VALUED_2:def 7 :: thesis: X is integer-functions-membered

let x be object ; :: according to VALUED_2:def 6 :: thesis: ( x in X implies x is INT -valued Function )

assume x in X ; :: thesis: x is INT -valued Function

then x is natural-valued Function by A1;

hence x is INT -valued Function ; :: thesis: verum