let A1, A2 be Subset of L; :: thesis: ( ( for x being Element of L holds

( x in A1 iff x is atom ) ) & ( for x being Element of L holds

( x in A2 iff x is atom ) ) implies A1 = A2 )

assume that

A2: for x being Element of L holds

( x in A1 iff x is atom ) and

A3: for x being Element of L holds

( x in A2 iff x is atom ) ; :: thesis: A1 = A2

for x being object holds

( x in A1 iff x in A2 ) by A2, A3;

hence A1 = A2 by TARSKI:2; :: thesis: verum

( x in A1 iff x is atom ) ) & ( for x being Element of L holds

( x in A2 iff x is atom ) ) implies A1 = A2 )

assume that

A2: for x being Element of L holds

( x in A1 iff x is atom ) and

A3: for x being Element of L holds

( x in A2 iff x is atom ) ; :: thesis: A1 = A2

for x being object holds

( x in A1 iff x in A2 ) by A2, A3;

hence A1 = A2 by TARSKI:2; :: thesis: verum