let X1, X2 be Subset of (C_VectorSpace_of_LinearOperators (X,Y)); :: thesis: ( ( for x being set holds
( x in X1 iff x is Lipschitzian LinearOperator of X,Y ) ) & ( for x being set holds
( x in X2 iff x is Lipschitzian LinearOperator of X,Y ) ) implies X1 = X2 )
assume that
A3: for x being set holds
( x in X1 iff x is Lipschitzian LinearOperator of X,Y ) and
A4: for x being set holds
( x in X2 iff x is Lipschitzian LinearOperator of X,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 Lipschitzian LinearOperator of X,Y by A4;
hence x in X1 by A3; :: thesis: verum

end;

then A5: X2 c= X1 by TARSKI:def 3;
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 Lipschitzian LinearOperator of X,Y by A3;
hence x in X2 by A4; :: thesis: verum

end;

then X1 c= X2 by TARSKI:def 3;
hence X1 = X2 by A5, XBOOLE_0:def 10; :: thesis: verum