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 is invertible
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 is invertible
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 is invertible
let g be Morphism of b,c; :: thesis: ( f is invertible & g is invertible implies g * f is invertible )
assume A3: f is invertible ; :: thesis: ( not g is invertible or g * f is invertible )
then A4: ( Hom (a,b) <> {} & Hom (b,a) <> {} ) by Dfi;
consider f1 being Morphism of b,a such that
A5: f * f1 = id b and
A6: f1 * f = id a by Dfi, A3;
assume A7: g is invertible ; :: thesis: g * f is invertible
then A8: ( Hom (b,c) <> {} & Hom (c,b) <> {} ) by Dfi;
consider g1 being Morphism of c,b such that
A9: g * g1 = id c and
A10: g1 * g = id b by Dfi, A7;

A11: now :: thesis: ( Hom (c,a) <> {} & ex f1g1 being Morphism of c,a st
( (g * f) * f1g1 = id c & f1g1 * (g * f) = id a ) )

thus A12: Hom (c,a) <> {} by A4, A8, Th23; :: thesis: ex f1g1 being Morphism of c,a st
( (g * f) * f1g1 = id c & f1g1 * (g * f) = id a )
take f1g1 = f1 * g1; :: thesis: ( (g * f) * f1g1 = id c & f1g1 * (g * f) = id a )
thus (g * f) * f1g1 = g * (f * (f1 * g1)) by A4, A8, A12, Th25
.= g * ((id b) * g1) by A4, A5, A8, Th25
.= id c by A8, A9, Th28 ; :: thesis: f1g1 * (g * f) = id a
Hom (a,c) <> {} by A4, A8, Th23;
hence f1g1 * (g * f) = f1 * (g1 * (g * f)) by A4, A8, Th25
.= f1 * ((id b) * f) by A4, A8, A10, Th25
.= id a by A4, A6, Th28 ;
:: thesis: verum

end;

Hom (a,c) <> {} by A4, A8, Th23;
hence g * f is invertible by A11, Th41; :: thesis: verum