reserve A,B,C for Category,
  F,F1 for Functor of A,B;
reserve o,m for set;
reserve t for natural_transformation of F,F1;

theorem Th31:
  for F1,F2,G1,G2 being Functor of A, [:B,C:] st F1
  is_naturally_transformable_to F2 & G1 is_naturally_transformable_to G2 for s
being natural_transformation of F1,F2, t being natural_transformation of G1,G2
  st Pr1 s = Pr1 t & Pr2 s = Pr2 t holds s = t
proof
  let F1,F2,G1,G2 be Functor of A, [:B,C:] such that
A1: F1 is_naturally_transformable_to F2 and
A2: G1 is_naturally_transformable_to G2;
  let s be natural_transformation of F1,F2, t be natural_transformation of G1,
  G2 such that
A3: Pr1 s = Pr1 t and
A4: Pr2 s = Pr2 t;
  reconsider S = s, T = t as Function of the carrier of A, [:the carrier' of B
  , the carrier' of C:];
A5: pr2(the carrier' of B, the carrier' of C)*S = pr2(B,C)*S
    .= Pr2 s by A1,Th28
    .= pr2(B,C)*T by A2,A4,Th28
    .= pr2(the carrier' of B, the carrier' of C)*T;
  pr1(the carrier' of B, the carrier' of C)*S = pr1(B,C)*S
    .= Pr1 s by A1,Th28
    .= pr1(B,C)*T by A2,A3,Th28
    .= pr1(the carrier' of B, the carrier' of C)*T;
  hence thesis by A5,FUNCT_3:80;
end;
