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 coretraction holds
f is one-to-one
let a, b be object of C; :: thesis: ( <^a,b^> <> {} & <^b,a^> <> {} implies for f being Morphism of a,b st f is coretraction holds
f is one-to-one )
assume A1: ( <^a,b^> <> {} & <^b,a^> <> {} ) ; :: thesis: for f being Morphism of a,b st f is coretraction holds
f is one-to-one
let f be Morphism of a,b; :: thesis: ( f is coretraction implies f is one-to-one )
given g being Morphism of b,a such that A2: g is_left_inverse_of f ; :: according to ALTCAT_3:def 3 :: thesis: f is one-to-one
A3: g * f = idm a by A2, ALTCAT_3:def 1;
A4: g * f = g * f by A1, Th38;
A5: dom f = the_carrier_of a by A1, Th36;
idm a = id (the_carrier_of a) by Def10;
hence f is one-to-one by A3, A4, A5, FUNCT_1:53; :: thesis: verum