A1: RealVectSpace (X,Y) = RLSStruct(# (Funcs (X, the carrier of Y)),(FuncZero (X,Y)),(FuncAdd (X,Y)),(FuncExtMult (X,Y)) #) by LOPBAN_1:def 4;
defpred S₁[ object ] means $1 is bounded Function of X, the carrier of Y;
consider IT being set such that
A2: for x being object holds
( x in IT iff ( x in Funcs (X, the carrier of Y) & S₁[x] ) ) from XBOOLE_0:sch 1();
take IT ; :: thesis: ( IT is Subset of (RealVectSpace (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 A2;
hence IT is Subset of (RealVectSpace (X,Y)) by A1, 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 A2; :: thesis: ( x is bounded Function of X, the carrier of Y implies x in IT )
assume A3: 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 A2, A3; :: thesis: verum