let A be ext-real-membered non empty non left_end non right_end interval set ; :: thesis: A = ].(inf A),(sup A).[
let x be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( not x in A or x in ].(inf A),(sup A).[ ) & ( not x in ].(inf A),(sup A).[ or x in A ) )
defpred S1[ ExtReal] means ( \$1 in A & \$1 < x );
defpred S2[ ExtReal] means ( \$1 in A & \$1 > x );
thus ( x in A implies x in ].(inf A),(sup A).[ ) :: thesis: ( not x in ].(inf A),(sup A).[ or x in A )
proof
assume A1: x in A ; :: thesis: x in ].(inf A),(sup A).[
then A2: x <> sup A by Def6;
x <= sup A by ;
then A3: x < sup A by ;
A4: x <> inf A by ;
inf A <= x by ;
then inf A < x by ;
hence x in ].(inf A),(sup A).[ by ; :: thesis: verum
end;
assume A5: x in ].(inf A),(sup A).[ ; :: thesis: x in A
per cases ( for r being ExtReal holds not S1[r] or for r being ExtReal holds not S2[r] or ( ex r being ExtReal st S1[r] & ex r being ExtReal st S2[r] ) ) ;
suppose for r being ExtReal holds not S1[r] ; :: thesis: x in A
end;
suppose for r being ExtReal holds not S2[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 that A6: ex r being ExtReal st S1[r] and
A7: ex r being ExtReal st S2[r] ; :: thesis: x in A
consider r being ExtReal such that
A8: r in A and
A9: r < x by A6;
consider s being ExtReal such that
A10: s in A and
A11: s > x by A7;
A12: x in [.r,s.] by ;
[.r,s.] c= A by ;
hence x in A by A12; :: thesis: verum
end;
end;