let C be category; :: thesis: for D being non empty subcategory of C
for o being Object of D
for o9 being Object of C st o = o9 holds
idm o = idm o9

let D be non empty subcategory of C; :: thesis: for o being Object of D
for o9 being Object of C st o = o9 holds
idm o = idm o9

let o be Object of D; :: thesis: for o9 being Object of C st o = o9 holds
idm o = idm o9

let o9 be Object of C; :: thesis: ( o = o9 implies idm o = idm o9 )
assume A1: o = o9 ; :: thesis: idm o = idm o9
then reconsider m = idm o9 as Morphism of o,o by Def14;
A2: idm o9 in <^o,o^> by ;
now :: thesis: for p being Object of D st <^o,p^> <> {} holds
for a being Morphism of o,p holds a * m = a
let p be Object of D; :: thesis: ( <^o,p^> <> {} implies for a being Morphism of o,p holds a * m = a )
assume A3: <^o,p^> <> {} ; :: thesis: for a being Morphism of o,p holds a * m = a
reconsider p9 = p as Object of C by Th29;
A4: <^o9,p9^> <> {} by ;
let a be Morphism of o,p; :: thesis: a * m = a
reconsider n = a as Morphism of o9,p9 by A1, A3, Th33;
thus a * m = n * (idm o9) by A1, A2, A3, Th32
.= a by ; :: thesis: verum
end;
hence idm o = idm o9 by ALTCAT_1:def 17; :: thesis: verum