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 S1[ 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 )
proof
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 ;
then x < sup A by ;
hence x in [.(min A),(sup A).[ by ; :: thesis: verum
end;
assume A4: x in [.(min A),(sup A).[ ; :: thesis: x in A
per cases ( for r being ExtReal holds not S1[r] or ex r being ExtReal st S1[r] ) ;
suppose for r being ExtReal 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 ; :: thesis: verum
end;
suppose ex r being ExtReal st S1[r] ; :: thesis: x in A
then consider r being ExtReal such that
A5: r in A and
A6: r > x ;
inf A <= x by ;
then A7: x in [.(inf A),r.] by ;
min A in A by Def5;
then [.(inf A),r.] c= A by ;
hence x in A by A7; :: thesis: verum
end;
end;