let X1, X2 be Subset of (ComplexVectSpace (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

A3: for x being set holds

( x in X1 iff x is bounded Function of X, the carrier of Y ) and

A4: 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

for x being object st x in X1 holds

x in X2

hence X1 = X2 by A5; :: thesis: verum

( 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

A3: for x being set holds

( x in X1 iff x is bounded Function of X, the carrier of Y ) and

A4: 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

then A5:
X2 c= X1
;
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 A4;

hence x in X1 by A3; :: thesis: verum

end;assume x in X2 ; :: thesis: x in X1

then x is bounded Function of X, the carrier of Y by A4;

hence x in X1 by A3; :: thesis: verum

for x being object st x in X1 holds

x in X2

proof

then
X1 c= X2
;
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 A3;

hence x in X2 by A4; :: thesis: verum

end;assume x in X1 ; :: thesis: x in X2

then x is bounded Function of X, the carrier of Y by A3;

hence x in X2 by A4; :: thesis: verum

hence X1 = X2 by A5; :: thesis: verum