reserve a,b,c,d,e for Real;

theorem Th07:
  for b,c,d being non negative Real st a <= b * c * d &
  d < e / (2 * b * |. c .|) holds a <= e / 2
  proof
    let b,c,d be non negative Real;
    assume that
A1: a <= b * c * d and
A2: d < e / (2 * b * |. c .|);
A3: 0 < b & 0 < c by A2,Th05;
A4: (b * c) * d <= (b * c) * (e / (2 * b * |. c .|)) by A2,XREAL_1:64;
    (b * c) * (e / (2 * b * |. c .|)) = (b * c) * (e / (2 * b * c))
      by ABSVALUE:def 1
                                     .= (b * c) * (e / (2 * (b * c)))
                                     .= e / 2 by A3,Th06;
    hence thesis by A4,A1,XXREAL_0:2;
  end;
