let X1, X2 be Subset of AS; :: thesis: ( ( for x being Element of AS holds

( x in X1 iff LIN a,b,x ) ) & ( for x being Element of AS holds

( x in X2 iff LIN a,b,x ) ) implies X1 = X2 )

assume that

A2: for x being Element of AS holds

( x in X1 iff LIN a,b,x ) and

A3: for x being Element of AS holds

( x in X2 iff LIN a,b,x ) ; :: thesis: X1 = X2

for x being object holds

( x in X1 iff x in X2 ) by A2, A3;

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

( x in X1 iff LIN a,b,x ) ) & ( for x being Element of AS holds

( x in X2 iff LIN a,b,x ) ) implies X1 = X2 )

assume that

A2: for x being Element of AS holds

( x in X1 iff LIN a,b,x ) and

A3: for x being Element of AS holds

( x in X2 iff LIN a,b,x ) ; :: thesis: X1 = X2

for x being object holds

( x in X1 iff x in X2 ) by A2, A3;

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