reserve Y for non empty set;

theorem th18:
  for a,b,c being Function of Y,BOOLEAN holds (c 'imp' a)=I_el(Y)
  & (c 'imp' b)=I_el(Y) implies c 'imp' (a '&' b)=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  assume that
A1: (c 'imp' a)=I_el(Y) and
A2: (c 'imp' b)=I_el(Y);
  for x being Element of Y holds (c 'imp' (a '&' b)).x=TRUE
  proof
    let x be Element of Y;
    (c 'imp' a).x= TRUE by A1,BVFUNC_1:def 11;
    then
A3: 'not' c.x 'or' a.x = TRUE by BVFUNC_1:def 8;
    (c 'imp' b).x= TRUE by A2,BVFUNC_1:def 11;
    then
A4: 'not' c.x 'or' b.x = TRUE by BVFUNC_1:def 8;
    (c 'imp' (a '&' b)).x ='not' c.x 'or' (a '&' b).x by BVFUNC_1:def 8
      .='not' c.x 'or' (a.x '&' b.x) by MARGREL1:def 20
      .=TRUE '&' TRUE by A3,A4,XBOOLEAN:9
      .=TRUE;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
