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
  All(a 'or' b,PA,G) '<' Ex(a,PA,G) 'or' Ex(b,PA,G)
proof
  let z be Element of Y;
A1: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  assume All(a 'or' b,PA,G).z=TRUE;
  then (a 'or' b).z=TRUE by A1,Lm1;
  then
A2: a.z 'or' b.z=TRUE by BVFUNC_1:def 4;
A3: a.z=TRUE or a.z=FALSE by XBOOLEAN:def 3;
  per cases by A2,A3,BINARITH:3;
  suppose
    a.z=TRUE;
    then B_SUP(a,CompF(PA,G)).z = TRUE by A1,BVFUNC_1:def 17;
    then Ex(a,PA,G).z=TRUE by BVFUNC_2:def 10;
    hence (Ex(a,PA,G) 'or' Ex(b,PA,G)).z =TRUE 'or' Ex(b,PA,G).z by
BVFUNC_1:def 4
      .=TRUE by BINARITH:10;
  end;
  suppose
    b.z=TRUE;
    then B_SUP(b,CompF(PA,G)).z = TRUE by A1,BVFUNC_1:def 17;
    then Ex(b,PA,G).z=TRUE by BVFUNC_2:def 10;
    hence (Ex(a,PA,G) 'or' Ex(b,PA,G)).z =Ex(a,PA,G).z 'or' TRUE by
BVFUNC_1:def 4
      .=TRUE by BINARITH:10;
  end;
end;
