reserve Y for non empty set,
  G for Subset of PARTITIONS(Y);

theorem
  for a,b being Function of Y,BOOLEAN, PA being a_partition of Y
  holds Ex(a,PA,G) 'xor' Ex(b,PA,G) '<' Ex(a 'xor' b,PA,G)
proof
  let a,b be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
  let z be Element of Y;
A1: (Ex(a,PA,G) 'xor' Ex(b,PA,G)).z =Ex(a,PA,G).z 'xor' Ex(b,PA,G).z by
BVFUNC_1:def 5
    .=('not' Ex(a,PA,G).z '&' Ex(b,PA,G).z) 'or' (Ex(a,PA,G).z '&' 'not' Ex(
  b,PA,G).z);
A2: ('not' Ex(a,PA,G).z '&' Ex(b,PA,G).z)=TRUE or ('not' Ex(a,PA,G).z '&' Ex
  (b,PA,G).z)=FALSE by XBOOLEAN:def 3;
A3: 'not' FALSE=TRUE by MARGREL1:11;
  assume
A4: (Ex(a,PA,G) 'xor' Ex(b,PA,G)).z=TRUE;
  now
    per cases by A4,A1,A2,BINARITH:3;
    case
A5:   ('not' Ex(a,PA,G).z '&' Ex(b,PA,G).z)=TRUE;
      then Ex(b,PA,G).z=TRUE by MARGREL1:12;
      then consider x1 being Element of Y such that
A6:   x1 in EqClass(z,CompF(PA,G)) and
A7:   b.x1=TRUE by BVFUNC_1:def 17;
      'not' Ex(a,PA,G).z=TRUE by A5,MARGREL1:12;
      then a.x1<>TRUE by A6,BVFUNC_1:def 17,MARGREL1:11;
      then
A8:   a.x1=FALSE by XBOOLEAN:def 3;
      (a 'xor' b).x1 =a.x1 'xor' b.x1 by BVFUNC_1:def 5
        .=TRUE 'or' FALSE by A3,A7,A8
        .=TRUE by BINARITH:10;
      hence thesis by A6,BVFUNC_1:def 17;
    end;
    case
A9:   (Ex(a,PA,G).z '&' 'not' Ex(b,PA,G).z)=TRUE;
      then Ex(a,PA,G).z=TRUE by MARGREL1:12;
      then consider x1 being Element of Y such that
A10:  x1 in EqClass(z,CompF(PA,G)) and
A11:  a.x1=TRUE by BVFUNC_1:def 17;
      'not' Ex(b,PA,G).z=TRUE by A9,MARGREL1:12;
      then b.x1<>TRUE by A10,BVFUNC_1:def 17,MARGREL1:11;
      then
A12:  b.x1=FALSE by XBOOLEAN:def 3;
      (a 'xor' b).x1=a.x1 'xor' b.x1 by BVFUNC_1:def 5
        .=FALSE 'or' TRUE by A3,A11,A12
        .=TRUE by BINARITH:10;
      hence thesis by A10,BVFUNC_1:def 17;
    end;
  end;
  hence thesis;
end;
