let C be Category; :: thesis: for D being Categorial Category
for I1 being Indexing of C
for I2 being Indexing of D
for T being TargetCat of I1 st D is TargetCat of I1 holds
I2 * I1 = I2 * (I1 -functor C,T)
let D be Categorial Category; :: thesis: for I1 being Indexing of C
for I2 being Indexing of D
for T being TargetCat of I1 st D is TargetCat of I1 holds
I2 * I1 = I2 * (I1 -functor C,T)
let I1 be Indexing of C; :: thesis: for I2 being Indexing of D
for T being TargetCat of I1 st D is TargetCat of I1 holds
I2 * I1 = I2 * (I1 -functor C,T)
let I2 be Indexing of D; :: thesis: for T being TargetCat of I1 st D is TargetCat of I1 holds
I2 * I1 = I2 * (I1 -functor C,T)
let T be TargetCat of I1; :: thesis: ( D is TargetCat of I1 implies I2 * I1 = I2 * (I1 -functor C,T) )
assume D is TargetCat of I1 ; :: thesis: I2 * I1 = I2 * (I1 -functor C,T)
then reconsider D' = D as TargetCat of I1 ;
A1: ( Image (I1 -functor C,D') = rng I1 & Image (I1 -functor C,T) = rng I1 & Image (I1 -functor C,(rng I1)) = rng I1 ) by Def12;
I1 -functor C,(rng I1) = I1 -functor C,T by Th11;
hence I2 * I1 = I2 * (I1 -functor C,T) by A1, Th22; :: thesis: verum