let Y1, Y2 be ext-real-membered set ; :: thesis: ( ( for x being ExtReal holds
( x in Y1 iff x is UpperBound of X ) ) & ( for x being ExtReal holds
( x in Y2 iff x is UpperBound of X ) ) implies Y1 = Y2 )
assume that
A2: for x being ExtReal holds
( x in Y1 iff x is UpperBound of X ) and
A3: for x being ExtReal holds
( x in Y2 iff x is UpperBound of X ) ; :: thesis: Y1 = Y2
let x be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( not x in Y1 or x in Y2 ) & ( not x in Y2 or x in Y1 ) )
( x in Y1 iff x is UpperBound of X ) by A2;
hence ( ( not x in Y1 or x in Y2 ) & ( not x in Y2 or x in Y1 ) ) by A3; :: thesis: verum