let X1, X2 be Subset of POS; :: thesis: ( ( for x being Element of POS holds
( x in X1 iff LIN a,b,x ) ) & ( for x being Element of POS holds
( x in X2 iff LIN a,b,x ) ) implies X1 = X2 )
assume that
A2: for x being Element of POS holds
( x in X1 iff LIN a,b,x ) and
A3: for x being Element of POS holds
( x in X2 iff LIN a,b,x ) ; :: thesis: X1 = X2
for x being set holds
( x in X1 iff x in X2 )

proof

let x be set ; :: thesis: ( x in X1 iff x in X2 )
thus ( x in X1 implies x in X2 ) :: thesis: ( x in X2 implies x in X1 )

proof

assume A4: x in X1 ; :: thesis: x in X2
then reconsider x9 = x as Element of POS ;
LIN a,b,x9 by A2, A4;
hence x in X2 by A3; :: thesis: verum

end;

assume A5: x in X2 ; :: thesis: x in X1
then reconsider x9 = x as Element of POS ;
LIN a,b,x9 by A3, A5;
hence x in X1 by A2; :: thesis: verum

end;

hence X1 = X2 by TARSKI:1; :: thesis: verum