let X1, X2 be Subset of (X *'); :: thesis: ( ( for x being set holds
( x in X1 iff x is Lipschitzian linear-Functional of X ) ) & ( for x being set holds
( x in X2 iff x is Lipschitzian linear-Functional of X ) ) implies X1 = X2 )

assume that
A3: for x being set holds
( x in X1 iff x is Lipschitzian linear-Functional of X ) and
A4: for x being set holds
( x in X2 iff x is Lipschitzian linear-Functional of X ) ; :: thesis: X1 = 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 Lipschitzian linear-Functional of X by A4;
hence x in X1 by A3; :: thesis: verum
end;
then A5: X2 c= X1 ;
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 Lipschitzian linear-Functional of X by A3;
hence x in X2 by A4; :: thesis: verum
end;
then X1 c= X2 ;
hence X1 = X2 by A5; :: thesis: verum