
theorem Th45:
  for A being category, a, b being Object of A
  for F1,F2 being Function st F1 in (the Arrows of Concretized A).(a,b) &
  F2 in (the Arrows of Concretized A).(a,b) &
  F1.[idm a, [a,a]] = F2.[idm a, [a,a]] holds F1 = F2
proof
  let A be category, a, b be Object of A;
  set B = Concretized A;
  let F1,F2 be Function such that
A1: F1 in (the Arrows of Concretized A).(a,b) and
A2: F2 in (the Arrows of Concretized A).(a,b) and
A3: F1.[idm a, [a,a]] = F2.[idm a, [a,a]];
A4: F1 in Funcs(B-carrier_of a, B-carrier_of b) by A1,Def12;
A5: F2 in Funcs(B-carrier_of a, B-carrier_of b) by A2,Def12;
A6: dom F1 = B-carrier_of a by A4,FUNCT_2:92;
A7: dom F2 = B-carrier_of a by A5,FUNCT_2:92;
  consider fa,fb being Object of A, f being Morphism of fa, fb such that
A8: fa = a and
A9: fb = b and
A10: <^fa, fb^> <> {} and
A11: for o being Object of A st <^o, fa^> <> {}
  for h being Morphism of o,fa holds F1.[h,[o,fa]] = [f*h,[o,fb]] by A1,Def12;
  consider ga,gb being Object of A, g being Morphism of ga, gb such that
A12: ga = a and
A13: gb = b and <^ga, gb^> <> {} and
A14: for o being Object of A st <^o, ga^> <> {}
  for h being Morphism of o,ga holds F2.[h,[o,ga]] = [g*h,[o,gb]] by A2,Def12;
  reconsider f, g as Morphism of a, b by A8,A9,A12,A13;
A15: F1.[idm a, [a,a]] = [f* idm a, [a,b]] by A8,A9,A11;
A16: f* idm a = f by A8,A9,A10,ALTCAT_1:def 17;
A17: g* idm a = g by A8,A9,A10,ALTCAT_1:def 17;
  F2.[idm a, [a,a]] = [g* idm a, [a,b]] by A12,A13,A14;
  then
A18: f = g by A3,A15,A16,A17,XTUPLE_0:1;
  now
    let x be object;
    assume x in B-carrier_of a;
    then consider bb being Object of A, ff being Morphism of bb,a such that
A19: <^bb,a^> <> {} and
A20: x = [ff,[bb,a]] by Th43;
    thus F1.x = [f*ff, [bb,b]] by A8,A9,A11,A19,A20
      .= F2.x by A12,A13,A14,A18,A19,A20;
  end;
  hence thesis by A6,A7;
end;
