reserve C for category,
  o1, o2, o3 for Object of C;

theorem
  AllEpi AllEpi C = AllEpi C
proof
A1: the carrier of AllEpi AllEpi C = the carrier of AllEpi C & the carrier
  of AllEpi C = the carrier of C by Def2;
A2: the Arrows of AllEpi AllEpi C cc= the Arrows of AllEpi C by Def2;
  now
    let i be object;
    assume
A3: i in [:the carrier of C,the carrier of C:];
    then consider o1, o2 being object such that
A4: o1 in the carrier of C & o2 in the carrier of C and
A5: i = [o1,o2] by ZFMISC_1:84;
    reconsider o1, o2 as Object of AllEpi C by A4,Def2;
    thus (the Arrows of AllEpi AllEpi C).i = (the Arrows of AllEpi C).i
    proof
      thus (the Arrows of AllEpi AllEpi C).i c= (the Arrows of AllEpi C).i by
A1,A2,A3;
      let n be object;
      assume
A6:   n in (the Arrows of AllEpi C).i;
      then reconsider n1 = n as Morphism of o1, o2 by A5;
      n1 is epi by A5,A6,Th51;
      then n in (the Arrows of AllEpi AllEpi C).(o1,o2) by A5,A6,Def2;
      hence thesis by A5;
    end;
  end;
  hence thesis by A1,ALTCAT_2:25,PBOOLE:3;
end;
