reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN, G being Subset of
PARTITIONS(Y), PA being a_partition of Y holds All(a 'eqv' b,PA,G) '<' All(a,PA
  ,G) 'eqv' All(b,PA,G)
proof
  let a,b be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA be a_partition of Y;
  let z be Element of Y;
  assume
A1: (All(a 'eqv' b,PA,G)).z=TRUE;
A2: (All(a,PA,G) 'eqv' All(b,PA,G)).z ='not'( All(a,PA,G).z 'xor' All(b,PA,G
  ).z) by BVFUNC_1:def 9
    .=((All(a,PA,G).z 'or' 'not' All(b,PA,G).z) '&' 'not' All(a,PA,G).z)
'or' ((All(a,PA,G).z 'or' 'not' All(b,PA,G).z) '&' All(b,PA,G).z) by XBOOLEAN:8
    .=('not' All(a,PA,G).z '&' All(a,PA,G).z 'or' 'not' All(a,PA,G).z '&'
'not' All(b,PA,G).z) 'or' (All(b,PA,G).z '&' (All(a,PA,G).z 'or' 'not' All(b,PA
  ,G).z)) by XBOOLEAN:8
    .=(('not' All(a,PA,G).z '&' All(a,PA,G).z) 'or' ('not' All(a,PA,G).z '&'
'not' All(b,PA,G).z)) 'or' ((All(b,PA,G).z '&' All(a,PA,G).z) 'or' (All(b,PA,G)
  .z '&' 'not' All(b,PA,G).z)) by XBOOLEAN:8
    .=(FALSE 'or' ('not' All(a,PA,G).z '&' 'not' All(b,PA,G).z)) 'or' ((All(
  b,PA,G).z '&' All(a,PA,G).z) 'or' (All(b,PA,G).z '&' 'not' All(b,PA,G).z))
by XBOOLEAN:138
    .=(('not' All(a,PA,G).z '&' 'not' All(b,PA,G).z)) 'or' ((All(b,PA,G).z
  '&' All(a,PA,G).z) 'or' FALSE) by XBOOLEAN:138
    .=('not' All(a,PA,G).z '&' 'not' All(b,PA,G).z) 'or' (All(b,PA,G).z '&'
  All(a,PA,G).z);
  per cases;
  suppose
A3: (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x
=TRUE) & for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x=TRUE
    ;
    then B_INF(b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
    hence thesis by A2,A3,BVFUNC_1:def 16;
  end;
  suppose
A4: (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a.x
=TRUE) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b.x
    =TRUE);
    then consider x1 being Element of Y such that
A5: x1 in EqClass(z,CompF(PA,G)) and
A6: b.x1<>TRUE;
A7: a.x1=TRUE by A4,A5;
    (a 'eqv' b).x1 ='not' (a.x1 'xor' b.x1) by BVFUNC_1:def 9
      .=FALSE by A6,A7,XBOOLEAN:def 3;
    hence thesis by A1,A5,BVFUNC_1:def 16;
  end;
  suppose
A8: not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds a.x=TRUE) & for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds
    b.x=TRUE;
    then consider x1 being Element of Y such that
A9: x1 in EqClass(z,CompF(PA,G)) and
A10: a.x1<>TRUE;
A11: b.x1=TRUE by A8,A9;
    (a 'eqv' b).x1 ='not' (a.x1 'xor' b.x1) by BVFUNC_1:def 9
      .=FALSE by A10,A11,XBOOLEAN:def 3;
    hence thesis by A1,A9,BVFUNC_1:def 16;
  end;
  suppose
A12: not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
holds a.x=TRUE) & not (for x being Element of Y st x in EqClass(z,CompF(PA,G))
    holds b.x=TRUE);
    then B_INF(b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
    hence thesis by A2,A12,BVFUNC_1:def 16;
  end;
end;
