let C be Category; :: thesis: for a, b, c being Object of C
for f being Morphism of a,b
for g being Morphism of b,c st f is invertible & g is invertible holds
(g * f) " = (f ") * (g ")
let a, b, c be Object of C; :: thesis: for f being Morphism of a,b
for g being Morphism of b,c st f is invertible & g is invertible holds
(g * f) " = (f ") * (g ")
let f be Morphism of a,b; :: thesis: for g being Morphism of b,c st f is invertible & g is invertible holds
(g * f) " = (f ") * (g ")
let g be Morphism of b,c; :: thesis: ( f is invertible & g is invertible implies (g * f) " = (f ") * (g ") )
assume that
A1: f is invertible and
A2: g is invertible ; :: thesis: (g * f) " = (f ") * (g ")
A3: Hom (a,b) <> {} by A1;
A4: Hom (b,c) <> {} by A2;
A5: Hom (c,b) <> {} by A2;
A6: Hom (b,a) <> {} by A1;
then A7: Hom (c,a) <> {} by A5, Th19;
then A8: ((f ") * (g ")) * (g * f) = (((f ") * (g ")) * g) * f by A3, A4, Th21
.= ((f ") * ((g ") * g)) * f by A4, A6, A5, Th21
.= ((f ") * (id b)) * f by A2, Def15
.= (f ") * f by A6, Th24
.= id a by A1, Def15 ;
A9: ( Hom (a,c) <> {} & g * f is invertible ) by A1, A2, Th19, Th40;
(g * f) * ((f ") * (g ")) = g * (f * ((f ") * (g "))) by A3, A4, A7, Th21
.= g * ((f * (f ")) * (g ")) by A3, A6, A5, Th21
.= g * ((id b) * (g ")) by A1, Def15
.= g * (g ") by A5, Th23
.= id c by A2, Def15 ;
hence (g * f) " = (f ") * (g ") by A8, A9, Def15; :: thesis: verum