let A, B be Category; :: thesis: for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds

for t1, t2 being transformation of F1,F2 st ( for a being Object of A holds t1 . a = t2 . a ) holds

t1 = t2

let F1, F2 be Functor of A,B; :: thesis: ( F1 is_transformable_to F2 implies for t1, t2 being transformation of F1,F2 st ( for a being Object of A holds t1 . a = t2 . a ) holds

t1 = t2 )

assume A1: F1 is_transformable_to F2 ; :: thesis: for t1, t2 being transformation of F1,F2 st ( for a being Object of A holds t1 . a = t2 . a ) holds

t1 = t2

let t1, t2 be transformation of F1,F2; :: thesis: ( ( for a being Object of A holds t1 . a = t2 . a ) implies t1 = t2 )

assume A2: for a being Object of A holds t1 . a = t2 . a ; :: thesis: t1 = t2

for t1, t2 being transformation of F1,F2 st ( for a being Object of A holds t1 . a = t2 . a ) holds

t1 = t2

let F1, F2 be Functor of A,B; :: thesis: ( F1 is_transformable_to F2 implies for t1, t2 being transformation of F1,F2 st ( for a being Object of A holds t1 . a = t2 . a ) holds

t1 = t2 )

assume A1: F1 is_transformable_to F2 ; :: thesis: for t1, t2 being transformation of F1,F2 st ( for a being Object of A holds t1 . a = t2 . a ) holds

t1 = t2

let t1, t2 be transformation of F1,F2; :: thesis: ( ( for a being Object of A holds t1 . a = t2 . a ) implies t1 = t2 )

assume A2: for a being Object of A holds t1 . a = t2 . a ; :: thesis: t1 = t2

now :: thesis: for a being Object of A holds t1 . a = t2 . a

hence
t1 = t2
; :: thesis: verumlet a be Object of A; :: thesis: t1 . a = t2 . a

thus t1 . a = t1 . a by A1, Def4

.= t2 . a by A2

.= t2 . a by A1, Def4 ; :: thesis: verum

end;thus t1 . a = t1 . a by A1, Def4

.= t2 . a by A2

.= t2 . a by A1, Def4 ; :: thesis: verum