reserve Y for non empty set;

theorem
  for a,u being Function of Y,BOOLEAN, G being Subset of
PARTITIONS(Y), PA being a_partition of Y st a 'imp' u = I_el(Y) holds All(a,PA,
  G) 'imp' u = I_el(Y)
proof
  let a,u be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA be a_partition of Y;
  assume
A1: a 'imp' u = I_el(Y);
  for x being Element of Y holds (All(a,PA,G) 'imp' u).x = TRUE
  proof
    let x be Element of Y;
    (a 'imp' u).x = TRUE by A1,BVFUNC_1:def 11;
    then
A2: 'not' a.x 'or' u.x = TRUE by BVFUNC_1:def 8;
A3: 'not' a.x=TRUE or 'not' a.x=FALSE by XBOOLEAN:def 3;
    now
      per cases by A2,A3;
      case
A4:     'not' a.x=TRUE;
        x in EqClass(x,CompF(PA,G)) by EQREL_1:def 6;
        then B_INF(a,CompF(PA,G)).x = FALSE by A4,BVFUNC_1:def 16;
        then (All(a,PA,G) 'imp' u).x =TRUE 'or' u.x by BVFUNC_1:def 8
          .=TRUE;
        hence thesis;
      end;
      case
        u.x=TRUE;
        then (All(a,PA,G) 'imp' u).x ='not' All(a,PA,G).x 'or' TRUE by
BVFUNC_1:def 8
          .=TRUE;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
