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

theorem Th3:
  t in ].r,s.[ implies |.t.| < max(|.r.|,|.s.|)
proof
  assume
A1: t in ].r,s.[;
  reconsider r,t,s as Real;
A2: r < t by A1,XXREAL_1:4;
A3: t < s by A1,XXREAL_1:4;
  per cases;
  suppose
A4: t >= 0;
    then t = |.t.| by ABSVALUE:def 1;
    then |.t.| < |.s.| by A3,A4,ABSVALUE:def 1;
    hence thesis by XXREAL_0:30;
  end;
  suppose
A5: t < 0;
    then
A6: -t = |.t.| by ABSVALUE:def 1;
    -r =|.r.| by A2,A5,ABSVALUE:def 1;
    then |.t.| < |.r.| by A2,A6,XREAL_1:24;
    hence thesis by XXREAL_0:30;
  end;
end;
