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 Ex(a,PA,G) 'imp' u '<' Ex(a 'imp' u,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 (Ex(a,PA,G) 'imp' u).z= TRUE;
  then
A2: ('not' Ex(a,PA,G).z) 'or' u.z=TRUE by BVFUNC_1:def 8;
A3: u.z= TRUE or u.z=FALSE by XBOOLEAN:def 3;
  now
    per cases by A2,A3,BINARITH:3;
    case
      ('not' Ex(a,PA,G).z)=TRUE;
      then
A4:   a.z<>TRUE by A1,BVFUNC_1:def 17,MARGREL1:11;
      (a 'imp' u).z =('not' a.z) 'or' u.z by BVFUNC_1:def 8
        .= TRUE 'or' u.z by A4,MARGREL1:11,XBOOLEAN:def 3
        .= TRUE by BINARITH:10;
      hence thesis by A1,BVFUNC_1:def 17;
    end;
    case
A5:   u.z=TRUE;
      (a 'imp' u).z =('not' a.z) 'or' u.z by BVFUNC_1:def 8
        .=TRUE by A5,BINARITH:10;
      hence thesis by A1,BVFUNC_1:def 17;
    end;
  end;
  hence thesis;
end;
