let A1, A2 be set ; :: thesis: ( ( for x being object holds

( x in A1 iff x is Probability of S ) ) & ( for x being object holds

( x in A2 iff x is Probability of S ) ) implies A1 = A2 )

assume that

A2: for x being object holds

( x in A1 iff x is Probability of S ) and

A3: for x being object holds

( x in A2 iff x is Probability of S ) ; :: thesis: A1 = A2

( x in A1 iff x is Probability of S ) ) & ( for x being object holds

( x in A2 iff x is Probability of S ) ) implies A1 = A2 )

assume that

A2: for x being object holds

( x in A1 iff x is Probability of S ) and

A3: for x being object holds

( x in A2 iff x is Probability of S ) ; :: thesis: A1 = A2

now :: thesis: for y being object holds

( y in A1 iff y in A2 )

hence
A1 = A2
by TARSKI:2; :: thesis: verum( y in A1 iff y in A2 )

let y be object ; :: 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;( y in A1 iff y is Probability of S ) by A2;

hence ( y in A1 iff y in A2 ) by A3; :: thesis: verum