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

theorem
  (a 'or' b) '<' (Ex(a,PA,G) 'or' Ex(b,PA,G))
proof
A1: Ex(a,PA,G) = B_SUP(a,CompF(PA,G)) by BVFUNC_2:def 10;
  let z be Element of Y;
A2: Ex(b,PA,G) = B_SUP(b,CompF(PA,G)) by BVFUNC_2:def 10;
A3: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume (a 'or' b).z=TRUE;
  then
A4: a.z 'or' b.z=TRUE by BVFUNC_1:def 4;
A5: b.z=TRUE or b.z=FALSE by XBOOLEAN:def 3;
  per cases by A4,A5,BINARITH:3;
  suppose
A6: a.z=TRUE;
    thus (Ex(a,PA,G) 'or' Ex(b,PA,G)).z =Ex(a,PA,G).z 'or' Ex(b,PA,G).z by
BVFUNC_1:def 4
      .=TRUE 'or' Ex(b,PA,G).z by A1,A3,A6,BVFUNC_1:def 17
      .=TRUE by BINARITH:10;
  end;
  suppose
A7: b.z=TRUE;
    thus (Ex(a,PA,G) 'or' Ex(b,PA,G)).z =Ex(a,PA,G).z 'or' Ex(b,PA,G).z by
BVFUNC_1:def 4
      .=Ex(a,PA,G).z 'or' TRUE by A2,A3,A7,BVFUNC_1:def 17
      .=TRUE by BINARITH:10;
  end;
end;
