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

theorem Th05:
  for b,c,d being non negative Real st
  d < e / (2 * b * |. c .|) holds
  b is positive & c is positive
  proof
    let b,c,d be non negative Real;
    assume
A1: d < e / (2 * b * |. c .|);
    assume not (b is positive & c is positive);
    then per cases;
    suppose b <= 0;
      then 2 * b * |.c.| = 0;
      hence contradiction by A1,XCMPLX_1:49;
    end;
    suppose c <= 0;
      then |. c .| = 0;
      hence contradiction by A1,XCMPLX_1:49;
    end;
  end;
