let X1, X2 be Subset of Q; :: thesis: ( ( for x being Element of Q holds
( x in X1 iff f . x = 1. Q2 ) ) & ( for x being Element of Q holds
( x in X2 iff f . x = 1. Q2 ) ) implies X1 = X2 )
assume A1: for x being Element of Q holds
( x in X1 iff f . x = 1. Q2 ) ; :: thesis: ( ex x being Element of Q st
( ( x in X2 implies f . x = 1. Q2 ) implies ( f . x = 1. Q2 & not x in X2 ) ) or X1 = X2 )
assume A2: for x being Element of Q holds
( x in X2 iff f . x = 1. Q2 ) ; :: thesis: X1 = X2
now :: thesis: for x being Element of Q holds
( x in X1 iff x in X2 )
let x be Element of Q; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff f . x = 1. Q2 ) by A1;
hence ( x in X1 iff x in X2 ) by A2; :: thesis: verum
end;
hence X1 = X2 by SUBSET_1:3; :: thesis: verum