
theorem Th30: :: JGRAPH_1:31
  for a,b,c,d being Real st a < c & c < b & a < d & d < b holds
  |.d - c qua Complex.| < b - a
proof
  let a,b,c,d be Real;
  assume that
A1: a < c and
A2: c < b & a < d and
A3: d < b;
A4: c + a < b + d by A2,XREAL_1:8;
  then c - d <= b - a by XREAL_1:21;
  then
A5: -(b - a) <= -(c - d) by XREAL_1:24;
A6: a + d < c + b by A1,A3,XREAL_1:8;
A7: |.d - c qua Complex.| <> b - a
  proof
    assume
A8: |.d - c qua Complex.| = b - a;
A9: d - c = b - a or d - c = -(b - a)
    proof
      per cases;
      suppose
        0 <= d - c;
        hence thesis by A8,ABSVALUE:def 1;
      end;
      suppose
        not 0 <= d - c;
        then b - a = -(d - c) by A8,ABSVALUE:def 1;
        hence thesis;
      end;
    end;
    now
      per cases by A9;
      case
        d - c = b - a;
        hence thesis by A6;
      end;
      case
        d - c = -(b - a);
        hence thesis by A4;
      end;
    end;
    hence thesis;
  end;
  d - c < b - a by A6,XREAL_1:21;
  then |.d - c qua Complex.| <= b - a by A5,ABSVALUE:5;
  hence thesis by A7,XXREAL_0:1;
end;
