reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds a '&' b '<' c
  implies a '<' (b 'imp' c)
proof
  let a,b,c be Function of Y,BOOLEAN;
  assume
A1: a '&' b '<' c;
  for x being Element of Y holds (a 'imp' (b 'imp' c)).x = TRUE
  proof
    let x be Element of Y;
A2: (a 'imp' (b 'imp' c)).x =('not' a.x) 'or' (b 'imp' c).x by BVFUNC_1:def 8
      .=('not' a.x) 'or' (('not' b.x) 'or' c.x) by BVFUNC_1:def 8
      .=('not' a.x 'or' 'not' b.x) 'or' c.x;
A3: (a '&' b 'imp' c).x =('not' (a '&' b).x) 'or' c.x by BVFUNC_1:def 8
      .=('not' a.x 'or' 'not' b.x) 'or' c.x by MARGREL1:def 20;
    (a '&' b) 'imp' c = I_el(Y) by A1,BVFUNC_1:16;
    hence thesis by A2,A3,BVFUNC_1:def 11;
  end;
  then a 'imp' (b 'imp' c) = I_el(Y) by BVFUNC_1:def 11;
  hence thesis by BVFUNC_1:16;
end;
