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

now :: thesis: for x being object st x in X1 holds
x in X2

let x be object ; :: thesis: ( x in X1 implies x in X2 )
assume x in X1 ; :: thesis: x in X2
then x is linear-Functional of X by A4;
hence x in X2 by A5; :: thesis: verum

end;

then A6: X1 c= X2 ;

now :: thesis: for x being object st x in X2 holds
x in X1

let x be object ; :: thesis: ( x in X2 implies x in X1 )
assume x in X2 ; :: thesis: x in X1
then x is linear-Functional of X by A5;
hence x in X1 by A4; :: thesis: verum

end;

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