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

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

( 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

then A5:
X2 c= X1
;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;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

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

x in X2

then
X1 c= X2
;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;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

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