let C be Category; :: thesis: for a, b being Object of C
for f being Morphism of a,b st Hom (a,b) <> {} & a is terminal holds
f is monic
let a, b be Object of C; :: thesis: for f being Morphism of a,b st Hom (a,b) <> {} & a is terminal holds
f is monic
let f be Morphism of a,b; :: thesis: ( Hom (a,b) <> {} & a is terminal implies f is monic )
assume that
A1: Hom (a,b) <> {} and
A2: a is terminal ; :: thesis: f is monic
now :: thesis: for c being Object of C
for g, h being Morphism of c,a st Hom (c,a) <> {} & f * g = f * h holds
g = h
let c be Object of C; :: thesis: for g, h being Morphism of c,a st Hom (c,a) <> {} & f * g = f * h holds
g = h
let g, h be Morphism of c,a; :: thesis: ( Hom (c,a) <> {} & f * g = f * h implies g = h )
assume that
Hom (c,a) <> {} and
f * g = f * h ; :: thesis: g = h
consider ff being Morphism of c,a such that
A3: for gg being Morphism of c,a holds ff = gg by A2;
ff = g by A3;
hence g = h by A3; :: thesis: verum
end;
hence f is monic by A1; :: thesis: verum