reserve r, s, t for Real;

theorem Th37: ::: SQUARE_1 or ABSVALUE
  r < s & s < t implies |.s.| < |.r.| + |.t.|
proof
  assume that
A1: r < s and
A2: s < t;
  per cases;
  suppose
A3: s<0;
    -s < -r by A1,XREAL_1:24; then
A4: -s+0 < -r+|.t.| by COMPLEX1:46,XREAL_1:8;
    -s = |.s.| by A3,ABSVALUE:def 1;
    hence thesis by A1,A3,A4,ABSVALUE:def 1;
  end;
  suppose
A5: 0<=s;
A6: s+0 < t+|.r.| by A2,COMPLEX1:46,XREAL_1:8;
    s = |.s.| by A5,ABSVALUE:def 1;
    hence thesis by A2,A5,A6,ABSVALUE:def 1;
  end;
end;
