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 (a 'eqv' b)=I_el(Y) implies (All
  (a,PA,G) 'eqv' All(b,PA,G))=I_el(Y)
proof
  let a,b be Function of Y,BOOLEAN;
  let G be Subset of PARTITIONS(Y);
  let PA be a_partition of Y;
  assume
A1: (a 'eqv' b)=I_el(Y);
  then (b 'imp' a)=I_el(Y) by Th10;
  then
A2: 'not' b 'or' a = I_el(Y) by Th8;
  (a 'imp' b)=I_el(Y) by A1,Th10;
  then
A3: 'not' a 'or' b = I_el(Y) by Th8;
  for z being Element of Y holds (All(a,PA,G) 'eqv' All(b,PA,G)).z=TRUE
  proof
    let z be Element of Y;
A4: now
      assume that
A5:   for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a
      .x=TRUE and
A6:   not(for x being Element of Y st x in EqClass(z,CompF(PA,G))
      holds b .x=TRUE);
      consider x1 being Element of Y such that
A7:   x1 in EqClass(z,CompF(PA,G)) and
A8:   b.x1<>TRUE by A6;
A9:   a.x1=TRUE by A5,A7;
A10:  b.x1=FALSE by A8,XBOOLEAN:def 3;
      ('not' a 'or' b).x1 =('not' a).x1 'or' b.x1 by BVFUNC_1:def 4
        .=FALSE 'or' FALSE by A10,A9,MARGREL1:def 19
        .=FALSE;
      hence thesis by A3,BVFUNC_1:def 11;
    end;
A11: now
      assume that
A12:  not(for x being Element of Y st x in EqClass(z,CompF(PA,G))
      holds a.x= TRUE) and
A13:  for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b .x= TRUE;
      consider x1 being Element of Y such that
A14:  x1 in EqClass(z,CompF(PA,G)) and
A15:  a.x1<>TRUE by A12;
A16:  b.x1=TRUE by A13,A14;
A17:  a.x1=FALSE by A15,XBOOLEAN:def 3;
      ('not' b 'or' a).x1 =('not' b).x1 'or' a.x1 by BVFUNC_1:def 4
        .=FALSE 'or' FALSE by A17,A16,MARGREL1:def 19
        .=FALSE;
      hence thesis by A2,BVFUNC_1:def 11;
    end;
    All(a,PA,G) 'eqv' All(b,PA,G) =(All(a,PA,G) 'imp' All(b,PA,G)) '&' (
    All(b,PA,G) 'imp' All(a,PA,G)) by Th7
      .=('not' All(a,PA,G) 'or' All(b,PA,G)) '&' (All(b,PA,G) 'imp' All(a,PA
    ,G)) by Th8
      .=('not' All(a,PA,G) 'or' All(b,PA,G)) '&' ('not' All(b,PA,G) 'or' All
    (a,PA,G)) by Th8
      .=(('not' All(a,PA,G) 'or' All(b,PA,G)) '&' 'not' All(b,PA,G)) 'or' ((
    'not' All(a,PA,G) 'or' All(b,PA,G)) '&' All(a,PA,G)) by BVFUNC_1:12
      .=(('not' All(a,PA,G) '&' 'not' All(b,PA,G)) 'or' (All(b,PA,G) '&'
'not' All(b,PA,G))) 'or' (('not' All(a,PA,G) 'or' All(b,PA,G)) '&' All(a,PA,G))
    by BVFUNC_1:12
      .=(('not' All(a,PA,G) '&' 'not' All(b,PA,G)) 'or' (All(b,PA,G) '&'
'not' All(b,PA,G))) 'or' (('not' All(a,PA,G) '&' All(a,PA,G)) 'or' (All(b,PA,G)
    '&' All(a,PA,G))) by BVFUNC_1:12
      .=(('not' All(a,PA,G) '&' 'not' All(b,PA,G)) 'or' O_el(Y)) 'or' ((
'not' All(a,PA,G) '&' All(a,PA,G)) 'or' (All(b,PA,G) '&' All(a,PA,G))) by Th5
      .=(('not' All(a,PA,G) '&' 'not' All(b,PA,G)) 'or' O_el(Y)) 'or' (O_el(
    Y) 'or' (All(b,PA,G) '&' All(a,PA,G))) by Th5
      .=('not' All(a,PA,G) '&' 'not' All(b,PA,G)) 'or' (O_el(Y) 'or' (All(b,
    PA,G) '&' All(a,PA,G))) by BVFUNC_1:9
      .=('not' All(a,PA,G) '&' 'not' All(b,PA,G)) 'or' (All(b,PA,G) '&' All(
    a,PA,G)) by BVFUNC_1:9;
    then
A18: (All(a,PA,G) 'eqv' All(b,PA,G)).z =('not' All(a,PA,G) '&' 'not' All(b,
    PA,G)).z 'or' ( All(b,PA,G) '&' All(a,PA,G)).z by BVFUNC_1:def 4
      .=('not' All(a,PA,G)).z '&' ('not' All(b,PA,G)).z 'or' ( All(b,PA,G)
    '&' All(a,PA,G)).z by MARGREL1:def 20
      .=(('not' All(a,PA,G)).z '&' ('not' All(b,PA,G)).z) 'or' All(b,PA,G).z
    '&' All(a,PA,G).z by MARGREL1:def 20
      .=('not' All(a,PA,G).z '&' ('not' All(b,PA,G)).z) 'or' ( All(b,PA,G).z
    '&' All(a,PA,G).z) by MARGREL1:def 19
      .=('not' All(a,PA,G).z '&' 'not' All(b,PA,G).z) 'or' ( All(b,PA,G).z
    '&' All(a,PA,G).z) by MARGREL1:def 19;
A19: now
      assume that
A20:  not(for x being Element of Y st x in EqClass(z,CompF(PA,G))
      holds a.x= TRUE) and
A21:  not(for x being Element of Y st x in EqClass(z,CompF(PA,G))
      holds b .x=TRUE);
      B_INF(b,CompF(PA,G)).z = FALSE by A21,BVFUNC_1:def 16;
      hence thesis by A18,A20,BVFUNC_1:def 16;
    end;
    now
      assume that
A22:  for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds a
      .x=TRUE and
A23:  for x being Element of Y st x in EqClass(z,CompF(PA,G)) holds b .x= TRUE;
      B_INF(b,CompF(PA,G)).z = TRUE by A23,BVFUNC_1:def 16;
      hence thesis by A18,A22,BVFUNC_1:def 16;
    end;
    hence thesis by A4,A11,A19;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
