defpred S_{1}[ object ] means $1 is bounded Function of X, the carrier of Y;

consider IT being set such that

A1: for x being object holds

( x in IT iff ( x in Funcs (X, the carrier of Y) & S_{1}[x] ) )
from XBOOLE_0:sch 1();

take IT ; :: thesis: ( IT is Subset of (ComplexVectSpace (X,Y)) & ( for x being set holds

( x in IT iff x is bounded Function of X, the carrier of Y ) ) )

for x being object st x in IT holds

x in Funcs (X, the carrier of Y) by A1;

hence IT is Subset of (ComplexVectSpace (X,Y)) by TARSKI:def 3; :: thesis: for x being set holds

( x in IT iff x is bounded Function of X, the carrier of Y )

let x be set ; :: thesis: ( x in IT iff x is bounded Function of X, the carrier of Y )

thus ( x in IT implies x is bounded Function of X, the carrier of Y ) by A1; :: thesis: ( x is bounded Function of X, the carrier of Y implies x in IT )

assume A2: x is bounded Function of X, the carrier of Y ; :: thesis: x in IT

then x in Funcs (X, the carrier of Y) by FUNCT_2:8;

hence x in IT by A1, A2; :: thesis: verum

consider IT being set such that

A1: for x being object holds

( x in IT iff ( x in Funcs (X, the carrier of Y) & S

take IT ; :: thesis: ( IT is Subset of (ComplexVectSpace (X,Y)) & ( for x being set holds

( x in IT iff x is bounded Function of X, the carrier of Y ) ) )

for x being object st x in IT holds

x in Funcs (X, the carrier of Y) by A1;

hence IT is Subset of (ComplexVectSpace (X,Y)) by TARSKI:def 3; :: thesis: for x being set holds

( x in IT iff x is bounded Function of X, the carrier of Y )

let x be set ; :: thesis: ( x in IT iff x is bounded Function of X, the carrier of Y )

thus ( x in IT implies x is bounded Function of X, the carrier of Y ) by A1; :: thesis: ( x is bounded Function of X, the carrier of Y implies x in IT )

assume A2: x is bounded Function of X, the carrier of Y ; :: thesis: x in IT

then x in Funcs (X, the carrier of Y) by FUNCT_2:8;

hence x in IT by A1, A2; :: thesis: verum