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

theorem Th32:
  for PA being a_partition of Y st u is_independent_of PA,G holds
  Ex(u 'or' a,PA,G) = u 'or' Ex(a,PA,G)
proof
  let PA be a_partition of Y;
  assume
A1: u is_independent_of PA,G;
A2: Ex(u 'or' a,PA,G) '<' u 'or' Ex(a,PA,G)
  proof
    let z be Element of Y;
A3: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
A4: (u 'or' Ex(a,PA,G)).z = u.z 'or' Ex(a,PA,G).z by BVFUNC_1:def 4;
    assume Ex(u 'or' a,PA,G).z= TRUE;
    then consider x1 being Element of Y such that
A5: x1 in EqClass(z,CompF(PA,G)) and
A6: (u 'or' a).x1=TRUE by BVFUNC_1:def 17;
A7: u.x1= TRUE or u.x1=FALSE by XBOOLEAN:def 3;
A8: u.x1 'or' a.x1=TRUE by A6,BVFUNC_1:def 4;
    now
      per cases by A8,A7,BINARITH:3;
      case
A9:     u.x1=TRUE;
        u.z=u.x1 by A1,A5,A3,BVFUNC_1:def 15;
        hence thesis by A4,A9,BINARITH:10;
      end;
      case
        a.x1=TRUE;
        then (u 'or' Ex(a,PA,G)).z = u.z 'or' TRUE by A5,A4,BVFUNC_1:def 17
          .= TRUE by BINARITH:10;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  u 'or' Ex(a,PA,G) '<' Ex(u 'or' a,PA,G)
  proof
    let z be Element of Y;
A10: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
    assume (u 'or' Ex(a,PA,G)).z= TRUE;
    then
A11: u.z 'or' Ex(a,PA,G).z=TRUE by BVFUNC_1:def 4;
A12: Ex(a,PA,G).z= TRUE or Ex(a,PA,G).z=FALSE by XBOOLEAN:def 3;
    now
      per cases by A11,A12,BINARITH:3;
      case
        u.z=TRUE;
        then (u 'or' a).z = TRUE 'or' a.z by BVFUNC_1:def 4
          .= TRUE by BINARITH:10;
        hence thesis by A10,BVFUNC_1:def 17;
      end;
      case
        Ex(a,PA,G).z=TRUE;
        then consider x1 being Element of Y such that
A13:    x1 in EqClass(z,CompF(PA,G)) and
A14:    a.x1=TRUE by BVFUNC_1:def 17;
        (u 'or' a).x1 = u.x1 'or' a.x1 by BVFUNC_1:def 4
          .= TRUE by A14,BINARITH:10;
        hence thesis by A13,BVFUNC_1:def 17;
      end;
    end;
    hence thesis;
  end;
  hence thesis by A2,BVFUNC_1:15;
end;
