let X1, X2 be set ; :: thesis: ( ( for x being object holds
( x in X1 iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) ) & ( for x being object holds
( x in X2 iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) ) implies X1 = X2 )
assume that
A6: for x being object holds
( x in X1 iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) and
A7: for x being object holds
( x in X2 iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) ; :: thesis: X1 = X2

now :: thesis: for x being object holds
( x in X1 iff x in X2 )

let x be object ; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) by A6;
hence ( x in X1 iff x in X2 ) by A7; :: thesis: verum

end;

hence X1 = X2 by TARSKI:2; :: thesis: verum