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(b 'imp' 'not' c,PA,G) '&' All(a 'imp' c,PA,G) '<' All(a 'imp'
  'not' b,PA,G)
proof
  let z be Element of Y;
  assume (All(b 'imp' 'not' c,PA,G) '&' All(a 'imp' c,PA,G)).z=TRUE;
  then
A1: All(b 'imp' 'not' c,PA,G).z '&' All(a 'imp' c,PA,G).z=TRUE by
MARGREL1:def 20;
A2: now
    assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds
    (a 'imp' c).x=TRUE);
    then B_INF(a 'imp' c,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
    then All(a 'imp' c,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence contradiction by A1,MARGREL1:12;
  end;
A3: now
    assume not (for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds
    (b 'imp' 'not' c).x=TRUE);
    then B_INF(b 'imp' 'not' c,CompF(PA,G)).z = FALSE by BVFUNC_1:def 16;
    then All(b 'imp' 'not' c,PA,G).z=FALSE by BVFUNC_2:def 9;
    hence contradiction by A1,MARGREL1:12;
  end;
  for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds (a 'imp'
  'not' b).x=TRUE
  proof
    let x be Element of Y;
A4: ('not' b.x)=TRUE or ('not' b.x)=FALSE by XBOOLEAN:def 3;
A5: ('not' a.x)=TRUE or ('not' a.x)=FALSE by XBOOLEAN:def 3;
    assume
A6: x in EqClass(z,CompF(PA,G));
    then (b 'imp' 'not' c).x=TRUE by A3;
    then
A7: ('not' b.x) 'or' ('not' c).x=TRUE by BVFUNC_1:def 8;
    (a 'imp' c).x=TRUE by A2,A6;
    then
A8: ('not' a.x) 'or' c.x=TRUE by BVFUNC_1:def 8;
    per cases by A7,A4,A8,A5,BINARITH:3;
    suppose
A9:   ('not' b.x)=TRUE & ('not' a.x)=TRUE;
      then ('not' b).x=TRUE by MARGREL1:def 19;
      hence (a 'imp' 'not' b).x =TRUE 'or' TRUE by A9,BVFUNC_1:def 8
        .=TRUE;
    end;
    suppose
      ('not' b.x)=TRUE & c.x=TRUE;
      then ('not' b).x=TRUE by MARGREL1:def 19;
      hence (a 'imp' 'not' b).x =('not' a.x) 'or' TRUE by BVFUNC_1:def 8
        .=TRUE by BINARITH:10;
    end;
    suppose
      ('not' c).x=TRUE & ('not' a.x)=TRUE;
      hence (a 'imp' 'not' b).x =TRUE 'or' ('not' b).x by BVFUNC_1:def 8
        .=TRUE by BINARITH:10;
    end;
    suppose
A10:  ('not' c).x=TRUE & c.x=TRUE;
      then 'not' c.x=TRUE by MARGREL1:def 19;
      hence thesis by A10,MARGREL1:11;
    end;
  end;
  then B_INF(a 'imp' 'not' b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 16;
  hence thesis by BVFUNC_2:def 9;
end;
