let X1, X2 be Subset of (RealVectSpace ( the carrier of [:X,Y:],Z)); :: thesis: ( ( for x being set holds
( x in X1 iff x is BilinearOperator of X,Y,Z ) ) & ( for x being set holds
( x in X2 iff x is BilinearOperator of X,Y,Z ) ) implies X1 = X2 )
assume that
A3: for x being set holds
( x in X1 iff x is BilinearOperator of X,Y,Z ) and
A4: for x being set holds
( x in X2 iff x is BilinearOperator of X,Y,Z ) ; :: thesis: X1 = X2
A5: X2 c= X1

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X2 or x in X1 )
assume x in X2 ; :: thesis: x in X1
then x is BilinearOperator of X,Y,Z by A4;
hence x in X1 by A3; :: thesis: verum

end;

X1 c= X2

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X1 or x in X2 )
assume x in X1 ; :: thesis: x in X2
then x is BilinearOperator of X,Y,Z by A3;
hence x in X2 by A4; :: thesis: verum

end;

hence X1 = X2 by A5, XBOOLE_0:def 10; :: thesis: verum