let A be ext-real-membered left_end non right_end interval set ; :: thesis: A = [.(min A),(sup A).[

let x be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( not x in A or x in [.(min A),(sup A).[ ) & ( not x in [.(min A),(sup A).[ or x in A ) )

defpred S_{1}[ ExtReal] means ( $1 in A & $1 > x );

thus ( x in A implies x in [.(min A),(sup A).[ ) :: thesis: ( not x in [.(min A),(sup A).[ or x in A )

let x be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( not x in A or x in [.(min A),(sup A).[ ) & ( not x in [.(min A),(sup A).[ or x in A ) )

defpred S

thus ( x in A implies x in [.(min A),(sup A).[ ) :: thesis: ( not x in [.(min A),(sup A).[ or x in A )

proof

assume A4:
x in [.(min A),(sup A).[
; :: thesis: x in A
A1:
not sup A in A
by Def6;

assume A2: x in A ; :: thesis: x in [.(min A),(sup A).[

then A3: min A <= x by Th3;

x <= sup A by A2, Th4;

then x < sup A by A2, A1, XXREAL_0:1;

hence x in [.(min A),(sup A).[ by A3, XXREAL_1:3; :: thesis: verum

end;assume A2: x in A ; :: thesis: x in [.(min A),(sup A).[

then A3: min A <= x by Th3;

x <= sup A by A2, Th4;

then x < sup A by A2, A1, XXREAL_0:1;

hence x in [.(min A),(sup A).[ by A3, XXREAL_1:3; :: thesis: verum

per cases
( for r being ExtReal holds not S_{1}[r] or ex r being ExtReal st S_{1}[r] )
;

end;

suppose
for r being ExtReal holds not S_{1}[r]
; :: thesis: x in A

then
x is UpperBound of A
by Def1;

then sup A <= x by Def3;

hence x in A by A4, XXREAL_1:3; :: thesis: verum

end;then sup A <= x by Def3;

hence x in A by A4, XXREAL_1:3; :: thesis: verum