reserve
  r,s,r0,s0,t for Real;

theorem Th1:
  for a,b being Real st r in [.a,b.] & s in [.a,b.] holds (r
  + s)/2 in [.a,b.]
proof
  let a,b be Real such that
A1: r in [.a,b.] and
A2: s in [.a,b.];
  reconsider a,b,r,s as Real;
A3: s <= b by A2,XXREAL_1:1;
  r <= b by A1,XXREAL_1:1;
  then r+s <= b+b by A3,XREAL_1:7;
  then
A4: (r+s)/2 <= (b+b)/2 by XREAL_1:72;
A5: a <= s by A2,XXREAL_1:1;
  a <= r by A1,XXREAL_1:1;
  then a+a <= r+s by A5,XREAL_1:7;
  then (a+a)/2 <= (r+s)/2 by XREAL_1:72;
  hence thesis by A4;
end;
