let A1, A2 be set ; :: thesis: ( ( for x being set holds
( x in A1 iff x is Probability of S ) ) & ( for x being set holds
( x in A2 iff x is Probability of S ) ) implies A1 = A2 )
assume that
A2: for x being set holds
( x in A1 iff x is Probability of S ) and
A3: for x being set holds
( x in A2 iff x is Probability of S ) ; :: thesis: A1 = A2

now

let y be set ; :: thesis: ( y in A1 iff y in A2 )
( y in A1 iff y is Probability of S ) by A2;
hence ( y in A1 iff y in A2 ) by A3; :: thesis: verum

end;

hence A1 = A2 by TARSKI:1; :: thesis: verum