reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds (a 'imp' c)=I_el(Y)
  & (b 'imp' c)=I_el(Y) implies (a 'or' b) 'imp' c = I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  assume that
A1: (a 'imp' c)=I_el(Y) and
A2: (b 'imp' c)=I_el(Y);
  for x being Element of Y holds ((a 'or' b) 'imp' c).x=TRUE
  proof
    let x be Element of Y;
    (a 'imp' c).x= TRUE by A1,BVFUNC_1:def 11;
    then
A3: 'not' a.x 'or' c.x = TRUE by BVFUNC_1:def 8;
    (b 'imp' c).x= TRUE by A2,BVFUNC_1:def 11;
    then
A4: 'not' b.x 'or' c.x = TRUE by BVFUNC_1:def 8;
    ((a 'or' b) 'imp' c).x ='not' (a 'or' b).x 'or' c.x by BVFUNC_1:def 8
      .='not'( a.x 'or' b.x) 'or' c.x by BVFUNC_1:def 4
      .=TRUE '&' TRUE by A3,A4,XBOOLEAN:9
      .=TRUE;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
