let I be set ; :: thesis: for C being Category
for c, d being Object of C
for F being Injections_family of c,I
for F9 being Injections_family of d,I st c is_a_coproduct_wrt F & d is_a_coproduct_wrt F9 & doms F = doms F9 holds
c,d are_isomorphic

let C be Category; :: thesis: for c, d being Object of C
for F being Injections_family of c,I
for F9 being Injections_family of d,I st c is_a_coproduct_wrt F & d is_a_coproduct_wrt F9 & doms F = doms F9 holds
c,d are_isomorphic

let c, d be Object of C; :: thesis: for F being Injections_family of c,I
for F9 being Injections_family of d,I st c is_a_coproduct_wrt F & d is_a_coproduct_wrt F9 & doms F = doms F9 holds
c,d are_isomorphic

let F be Injections_family of c,I; :: thesis: for F9 being Injections_family of d,I st c is_a_coproduct_wrt F & d is_a_coproduct_wrt F9 & doms F = doms F9 holds
c,d are_isomorphic

let F9 be Injections_family of d,I; :: thesis: ( c is_a_coproduct_wrt F & d is_a_coproduct_wrt F9 & doms F = doms F9 implies c,d are_isomorphic )
assume that
A1: c is_a_coproduct_wrt F and
A2: d is_a_coproduct_wrt F9 and
A3: doms F = doms F9 ; :: thesis:
consider fg being Morphism of C such that
fg in Hom (d,d) and
A4: for k being Morphism of C st k in Hom (d,d) holds
( ( for x being set st x in I holds
k (*) (F9 /. x) = F9 /. x ) iff fg = k ) by A2;
consider f being Morphism of C such that
A5: f in Hom (c,d) and
A6: for k being Morphism of C st k in Hom (c,d) holds
( ( for x being set st x in I holds
k (*) (F /. x) = F9 /. x ) iff f = k ) by A1, A3;
reconsider f = f as Morphism of c,d by ;
A7: dom f = c by ;
A8: now :: thesis: ( id c in Hom (c,c) & ( for x being set st x in I holds
(id c) (*) (F /. x) = F /. x ) )
set k = id c;
thus id c in Hom (c,c) by CAT_1:27; :: thesis: for x being set st x in I holds
(id c) (*) (F /. x) = F /. x

let x be set ; :: thesis: ( x in I implies (id c) (*) (F /. x) = F /. x )
assume x in I ; :: thesis: (id c) (*) (F /. x) = F /. x
then cod (F /. x) = c by Th62;
hence (id c) (*) (F /. x) = F /. x by CAT_1:21; :: thesis: verum
end;
A9: now :: thesis: ( id d in Hom (d,d) & ( for x being set st x in I holds
(id d) (*) (F9 /. x) = F9 /. x ) )
set k = id d;
thus id d in Hom (d,d) by CAT_1:27; :: thesis: for x being set st x in I holds
(id d) (*) (F9 /. x) = F9 /. x

let x be set ; :: thesis: ( x in I implies (id d) (*) (F9 /. x) = F9 /. x )
assume x in I ; :: thesis: (id d) (*) (F9 /. x) = F9 /. x
then cod (F9 /. x) = d by Th62;
hence (id d) (*) (F9 /. x) = F9 /. x by CAT_1:21; :: thesis: verum
end;
consider gf being Morphism of C such that
gf in Hom (c,c) and
A10: for k being Morphism of C st k in Hom (c,c) holds
( ( for x being set st x in I holds
k (*) (F /. x) = F /. x ) iff gf = k ) by A1;
consider g being Morphism of C such that
A11: g in Hom (d,c) and
A12: for k being Morphism of C st k in Hom (d,c) holds
( ( for x being set st x in I holds
k (*) (F9 /. x) = F /. x ) iff g = k ) by A2, A3;
reconsider g = g as Morphism of d,c by ;
take f ; :: according to CAT_1:def 20 :: thesis: f is invertible
thus ( Hom (c,d) <> {} & Hom (d,c) <> {} ) by ; :: according to CAT_1:def 16 :: thesis: ex b1 being Morphism of d,c st
( f * b1 = id d & b1 * f = id c )

take g ; :: thesis: ( f * g = id d & g * f = id c )
A13: cod f = d by ;
A14: dom g = d by ;
A15: cod g = c by ;
A16: now :: thesis: ( f (*) g in Hom (d,d) & ( for x being set st x in I holds
(f (*) g) (*) (F9 /. x) = F9 /. x ) )
( cod (f (*) g) = d & dom (f (*) g) = d ) by ;
hence f (*) g in Hom (d,d) ; :: thesis: for x being set st x in I holds
(f (*) g) (*) (F9 /. x) = F9 /. x

let x be set ; :: thesis: ( x in I implies (f (*) g) (*) (F9 /. x) = F9 /. x )
assume A17: x in I ; :: thesis: (f (*) g) (*) (F9 /. x) = F9 /. x
then cod (F9 /. x) = d by Th62;
hence (f (*) g) (*) (F9 /. x) = f (*) (g (*) (F9 /. x)) by
.= f (*) (F /. x) by
.= F9 /. x by A5, A6, A17 ;
:: thesis: verum
end;
thus f * g = f (*) g by
.= fg by
.= id d by A4, A9 ; :: thesis: g * f = id c
A18: now :: thesis: ( g (*) f in Hom (c,c) & ( for x being set st x in I holds
(g (*) f) (*) (F /. x) = F /. x ) )
( cod (g (*) f) = c & dom (g (*) f) = c ) by ;
hence g (*) f in Hom (c,c) ; :: thesis: for x being set st x in I holds
(g (*) f) (*) (F /. x) = F /. x

let x be set ; :: thesis: ( x in I implies (g (*) f) (*) (F /. x) = F /. x )
assume A19: x in I ; :: thesis: (g (*) f) (*) (F /. x) = F /. x
then cod (F /. x) = c by Th62;
hence (g (*) f) (*) (F /. x) = g (*) (f (*) (F /. x)) by
.= g (*) (F9 /. x) by A5, A6, A19
.= F /. x by ;
:: thesis: verum
end;
thus g * f = g (*) f by
.= gf by
.= id c by ; :: thesis: verum