reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);
reserve a, u for Function of Y,BOOLEAN;

theorem
  for PA being a_partition of Y holds u 'imp' Ex(a,PA,G) '<' Ex(u 'imp' a,PA,G)
proof
  let PA be a_partition of Y;
  let z be Element of Y;
A1: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume (u 'imp' Ex(a,PA,G)).z= TRUE;
  then
A2: ('not' u.z) 'or' Ex(a,PA,G).z=TRUE by BVFUNC_1:def 8;
A3: Ex(a,PA,G).z= TRUE or Ex(a,PA,G).z=FALSE by XBOOLEAN:def 3;
  now
    per cases by A2,A3,BINARITH:3;
    case
A4:   'not' u.z=TRUE;
      (u 'imp' a).z = ('not' u.z) 'or' a.z by BVFUNC_1:def 8
        .= TRUE by A4,BINARITH:10;
      hence thesis by A1,BVFUNC_1:def 17;
    end;
    case
      Ex(a,PA,G).z=TRUE;
      then consider x1 being Element of Y such that
A5:   x1 in EqClass(z,CompF(PA,G)) and
A6:   a.x1=TRUE by BVFUNC_1:def 17;
      (u 'imp' a).x1 =('not' u.x1) 'or' a.x1 by BVFUNC_1:def 8
        .=TRUE by A6,BINARITH:10;
      hence thesis by A5,BVFUNC_1:def 17;
    end;
  end;
  hence thesis;
end;
