reserve p, q, x, y for Real,
  n for Nat;

theorem Th10:
  for f being Element of REAL n st 0 <= x & x <= 1 holds |.x*f.| <= |.f.|
proof
  let f be Element of REAL n such that
A1: 0 <= x and
A2: x <= 1;
  |.x*f.| = |.x.|*|.f.| & x = |.x.| by A1,ABSVALUE:def 1,EUCLID:11;
  then |.x*f.| <= 1*|.f.| by A1,A2,XREAL_1:66;
  hence thesis;
end;
