let C be concrete category; :: thesis: for a, b being Object of C st <^a,b^> <> {} & <^b,a^> <> {} holds
for f being Morphism of a,b st f is retraction holds
rng f = the_carrier_of b
let a, b be Object of C; :: thesis: ( <^a,b^> <> {} & <^b,a^> <> {} implies for f being Morphism of a,b st f is retraction holds
rng f = the_carrier_of b )
assume that
A1: <^a,b^> <> {} and
A2: <^b,a^> <> {} ; :: thesis: for f being Morphism of a,b st f is retraction holds
rng f = the_carrier_of b
let f be Morphism of a,b; :: thesis: ( f is retraction implies rng f = the_carrier_of b )
given g being Morphism of b,a such that A3: g is_right_inverse_of f ; :: according to ALTCAT_3:def 2 :: thesis: rng f = the_carrier_of b
A4: f * g = idm b by A3;
A5: f * g = f * g by A1, A2, Th36;
A6: f is Function of (the_carrier_of a),(the_carrier_of b) by A1, Th34;
A7: g is Function of (the_carrier_of b),(the_carrier_of a) by A2, Th34;
idm b = id (the_carrier_of b) by Def10;
hence rng f = the_carrier_of b by A4, A5, A6, A7, FUNCT_2:18; :: thesis: verum