let X1, X2 be Subset of (RealVectSpace (X,Y)); :: thesis: ( ( for x being set holds
( x in X1 iff x is bounded Function of X, the carrier of Y ) ) & ( for x being set holds
( x in X2 iff x is bounded Function of X, the carrier of Y ) ) implies X1 = X2 )
assume that
A4: for x being set holds
( x in X1 iff x is bounded Function of X, the carrier of Y ) and
A5: for x being set holds
( x in X2 iff x is bounded Function of X, the carrier of Y ) ; :: thesis: X1 = X2
for x being object st x in X2 holds
x in X1

proof

let x be object ; :: thesis: ( x in X2 implies x in X1 )
assume x in X2 ; :: thesis: x in X1
then x is bounded Function of X, the carrier of Y by A5;
hence x in X1 by A4; :: thesis: verum

end;

then A6: X2 c= X1 ;
for x being object st x in X1 holds
x in X2

proof

let x be object ; :: thesis: ( x in X1 implies x in X2 )
assume x in X1 ; :: thesis: x in X2
then x is bounded Function of X, the carrier of Y by A4;
hence x in X2 by A5; :: thesis: verum

end;

then X1 c= X2 ;
hence X1 = X2 by A6; :: thesis: verum