let Y1, Y2 be ext-real-membered set ; :: thesis: ( ( for x being ExtReal holds

( x in Y1 iff x is LowerBound of X ) ) & ( for x being ExtReal holds

( x in Y2 iff x is LowerBound of X ) ) implies Y1 = Y2 )

assume that

A2: for x being ExtReal holds

( x in Y1 iff x is LowerBound of X ) and

A3: for x being ExtReal holds

( x in Y2 iff x is LowerBound 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 LowerBound 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

( x in Y1 iff x is LowerBound of X ) ) & ( for x being ExtReal holds

( x in Y2 iff x is LowerBound of X ) ) implies Y1 = Y2 )

assume that

A2: for x being ExtReal holds

( x in Y1 iff x is LowerBound of X ) and

A3: for x being ExtReal holds

( x in Y2 iff x is LowerBound 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 LowerBound 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