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

( 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

X1 c= X2
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;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

proof

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