let A be ext-real-membered left_end non right_end connected set ; :: thesis: A = [.(min A),(sup A).[
let x be ext-real number ; :: 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 ) )
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 Z: x in A ; :: thesis: x in [.(min A),(sup A).[
then A: ( min A <= x & x <= sup A ) by Th3, Th3B;
not sup A in A by Def6;
then x < sup A by A, Z, XXREAL_0:1;
hence x in [.(min A),(sup A).[ by A, XXREAL_1:3; :: thesis: verum

end;

assume Z: x in [.(min A),(sup A).[ ; :: thesis: x in A
defpred S₁[ ext-real number ] means ( $1 in A & $1 > x );

per cases ( for r being ext-real number holds not S₁[r] or ex r being ext-real number st S₁[r] ) ;

suppose for r being ext-real number holds not S₁[r] ; :: thesis: x in A

then x is UpperBound of A by Def1;
then sup A <= x by Def3;
hence x in A by Z, XXREAL_1:3; :: thesis: verum

end;

suppose ex r being ext-real number st S₁[r] ; :: thesis: x in A

then consider r being ext-real number such that
W3: r in A and
W4: r > x ;
B: min A in A by Def5;
inf A <= x by Z, XXREAL_1:3;
then A: x in [.(inf A),r.] by W4, XXREAL_1:1;
[.(inf A),r.] c= A by W3, B, Def12;
hence x in A by A; :: thesis: verum

end;

end;