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
  Ex(a,PA,G) 'imp' Ex(b,PA,G) '<' All(a,PA,G) 'imp' 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: ('not' Ex(a,PA,G).z)=TRUE or ('not' Ex(a,PA,G).z)=FALSE by XBOOLEAN:def 3;
  assume (Ex(a,PA,G) 'imp' Ex(b,PA,G)).z=TRUE;
  then
A3: ('not' Ex(a,PA,G).z) 'or' Ex(b,PA,G).z=TRUE by BVFUNC_1:def 8;
A4: z in EqClass(z,CompF(PA,G)) by EQREL_1:def 6;
  per cases by A3,A2,BINARITH:3;
  suppose
    ('not' Ex(a,PA,G).z)=TRUE;
    then Ex(a,PA,G).z=FALSE by MARGREL1:11;
    then a.z<>TRUE by A1,A4,BVFUNC_1:def 17;
    then B_INF(a,CompF(PA,G)).z = FALSE by A4,BVFUNC_1:def 16;
    then All(a,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence (All(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' FALSE) 'or' Ex(b,PA,G).z by
BVFUNC_1:def 8
      .=TRUE 'or' Ex(b,PA,G).z by MARGREL1:11
      .=TRUE by BINARITH:10;
  end;
  suppose
    Ex(b,PA,G).z=TRUE;
    hence (All(a,PA,G) 'imp' Ex(b,PA,G)).z =('not' All(a,PA,G).z) 'or' TRUE by
BVFUNC_1:def 8
      .=TRUE by BINARITH:10;
  end;
end;
