let r, s be real number ; :: thesis: ( r < s implies upper_bound [.r,s.[ = s )
set X = [.r,s.[;
assume A1: r < s ; :: thesis: upper_bound [.r,s.[ = s
A2: for a being real number st a in [.r,s.[ holds
a <= s by XXREAL_1:3;
reconsider r = r, s = s as Real by XREAL_0:def 1;
A3: r < (r + s) / 2 by A1, XREAL_1:228;
A4: (r + s) / 2 < s by A1, XREAL_1:228;
A5: for b being real number st 0 < b holds
ex a being real number st
( a in [.r,s.[ & s - b < a )

proof

let b be real number ; :: thesis: ( 0 < b implies ex a being real number st
( a in [.r,s.[ & s - b < a ) )
assume that
A6: 0 < b and
A7: for a being real number st a in [.r,s.[ holds
a <= s - b ; :: thesis: contradiction
reconsider b = b as Real by XREAL_0:def 1;

per cases ( s - b <= r or s - b > r ) ;

suppose s - b <= r ; :: thesis: contradiction

then A8: (r + s) / 2 > s - b by A3, XXREAL_0:2;
(r + s) / 2 in [.r,s.[ by A3, A4, XXREAL_1:3;
hence contradiction by A7, A8; :: thesis: verum

end;

suppose A9: s - b > r ; :: thesis: contradiction

A10: s - b < s - 0 by A6, XREAL_1:17;
then s - b < (s + (s - b)) / 2 by XREAL_1:228;
then A11: r < (s + (s - b)) / 2 by A9, XXREAL_0:2;
(s + (s - b)) / 2 < s by A10, XREAL_1:228;
then (s + (s - b)) / 2 in [.r,s.[ by A11, XXREAL_1:3;
hence contradiction by A7, A10, XREAL_1:228; :: thesis: verum

end;

end;

end;

not [.r,s.[ is empty by A1, XXREAL_1:3;
hence upper_bound [.r,s.[ = s by A2, A5, SEQ_4:def 4; :: thesis: verum