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 (Ex(
  a,PA,G) 'eqv' Ex(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 (Ex(a,PA,G) 'eqv' Ex(b,PA,G)).z=TRUE
  proof
    let z be Element of Y;
    Ex(a,PA,G) 'eqv' Ex(b,PA,G) =(Ex(a,PA,G) 'imp' Ex(b,PA,G)) '&' (Ex(b,
    PA,G) 'imp' Ex(a,PA,G)) by Th7
      .=('not' Ex(a,PA,G) 'or' Ex(b,PA,G)) '&' (Ex(b,PA,G) 'imp' Ex(a,PA,G))
    by Th8
      .=('not' Ex(a,PA,G) 'or' Ex(b,PA,G)) '&' ('not' Ex(b,PA,G) 'or' Ex(a,
    PA,G)) by Th8
      .=(('not' Ex(a,PA,G) 'or' Ex(b,PA,G)) '&' 'not' Ex(b,PA,G)) 'or' ((
    'not' Ex(a,PA,G) 'or' Ex(b,PA,G)) '&' Ex(a,PA,G)) by BVFUNC_1:12
      .=(('not' Ex(a,PA,G) '&' 'not' Ex(b,PA,G)) 'or' (Ex(b,PA,G) '&' 'not'
    Ex(b,PA,G))) 'or' (('not' Ex(a,PA,G) 'or' Ex(b,PA,G)) '&' Ex(a,PA,G)) by
BVFUNC_1:12
      .=(('not' Ex(a,PA,G) '&' 'not' Ex(b,PA,G)) 'or' (Ex(b,PA,G) '&' 'not'
Ex(b,PA,G))) 'or' (('not' Ex(a,PA,G) '&' Ex(a,PA,G)) 'or' (Ex(b,PA,G) '&' Ex(a,
    PA,G))) by BVFUNC_1:12
      .=(('not' Ex(a,PA,G) '&' 'not' Ex(b,PA,G)) 'or' O_el(Y)) 'or' (('not'
    Ex(a,PA,G) '&' Ex(a,PA,G)) 'or' (Ex(b,PA,G) '&' Ex(a,PA,G))) by Th5
      .=(('not' Ex(a,PA,G) '&' 'not' Ex(b,PA,G)) 'or' O_el(Y)) 'or' (O_el(Y)
    'or' (Ex(b,PA,G) '&' Ex(a,PA,G))) by Th5
      .=('not' Ex(a,PA,G) '&' 'not' Ex(b,PA,G)) 'or' (O_el(Y) 'or' (Ex(b,PA,
    G) '&' Ex(a,PA,G))) by BVFUNC_1:9
      .=('not' Ex(a,PA,G) '&' 'not' Ex(b,PA,G)) 'or' (Ex(b,PA,G) '&' Ex(a,PA
    ,G)) by BVFUNC_1:9;
    then
A4: (Ex(a,PA,G) 'eqv' Ex(b,PA,G)).z =('not' Ex(a,PA,G) '&' 'not' Ex(b,PA,G
    )).z 'or' ( Ex(b,PA,G) '&' Ex(a,PA,G)).z by BVFUNC_1:def 4
      .= ('not' Ex(a,PA,G)).z '&' ('not' Ex(b,PA,G)).z 'or' ( Ex(b,PA,G) '&'
    Ex(a,PA,G)).z by MARGREL1:def 20
      .= ('not' Ex(a,PA,G)).z '&' ('not' Ex(b,PA,G)).z 'or' (Ex(b,PA,G).z
    '&' Ex(a,PA,G).z) by MARGREL1:def 20
      .=('not' Ex(a,PA,G).z '&' ('not' Ex(b,PA,G)).z) 'or' ( Ex(b,PA,G).z
    '&' Ex(a,PA,G).z) by MARGREL1:def 19
      .=('not' Ex(a,PA,G).z '&' 'not' Ex(b,PA,G).z) 'or' ( Ex(b,PA,G).z '&'
    Ex(a,PA,G).z) by MARGREL1:def 19;
    per cases;
    suppose
A5:   (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x=
TRUE) & ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=TRUE;
      then B_SUP(b,CompF(PA,G)).z = TRUE by BVFUNC_1:def 17;
      hence thesis by A4,A5,BVFUNC_1:def 17;
    end;
    suppose
A6:   (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a.x=
TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=TRUE)
      ;
      then consider x1 being Element of Y such that
A7:   x1 in EqClass(z,CompF(PA,G)) and
A8:   a.x1=TRUE;
      b.x1<>TRUE by A6,A7;
      then
A9:   b.x1=FALSE by XBOOLEAN:def 3;
      ('not' a 'or' b).x1 =('not' a).x1 'or' b.x1 by BVFUNC_1:def 4
        .=FALSE 'or' FALSE by A8,A9,MARGREL1:def 19
        .=FALSE;
      hence thesis by A3,BVFUNC_1:def 11;
    end;
    suppose
A10:  not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a
.x=TRUE) & ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=TRUE;
      then consider x1 being Element of Y such that
A11:  x1 in EqClass(z,CompF(PA,G)) and
A12:  b.x1=TRUE;
      a.x1<>TRUE by A10,A11;
      then
A13:  a.x1=FALSE by XBOOLEAN:def 3;
      ('not' b 'or' a).x1 =('not' b).x1 'or' a.x1 by BVFUNC_1:def 4
        .=FALSE 'or' FALSE by A12,A13,MARGREL1:def 19
        .=FALSE;
      hence thesis by A2,BVFUNC_1:def 11;
    end;
    suppose
A14:  not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & a
.x=TRUE) & not (ex x being Element of Y st x in EqClass(z,CompF(PA,G)) & b.x=
      TRUE);
      then B_SUP(b,CompF(PA,G)).z = FALSE by BVFUNC_1:def 17;
      hence thesis by A4,A14,BVFUNC_1:def 17;
    end;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
