let C be Category; :: thesis: for a, b being Object of C

for f being Morphism of a,b st f is retraction holds

f is epi

let a, b be Object of C; :: thesis: for f being Morphism of a,b st f is retraction holds

f is epi

let f be Morphism of a,b; :: thesis: ( f is retraction implies f is epi )

assume A1: ( Hom (a,b) <> {} & Hom (b,a) <> {} ) ; :: according to CAT_3:def 8 :: thesis: ( for g being Morphism of b,a holds not f * g = id b or f is epi )

given g being Morphism of b,a such that A2: f * g = id b ; :: thesis: f is epi

thus Hom (a,b) <> {} by A1; :: according to CAT_1:def 15 :: thesis: for b_{1} being Element of the carrier of C holds

( Hom (b,b_{1}) = {} or for b_{2}, b_{3} being Morphism of b,b_{1} holds

( not b_{2} * f = b_{3} * f or b_{2} = b_{3} ) )

let c be Object of C; :: thesis: ( Hom (b,c) = {} or for b_{1}, b_{2} being Morphism of b,c holds

( not b_{1} * f = b_{2} * f or b_{1} = b_{2} ) )

assume A3: Hom (b,c) <> {} ; :: thesis: for b_{1}, b_{2} being Morphism of b,c holds

( not b_{1} * f = b_{2} * f or b_{1} = b_{2} )

let p1, p2 be Morphism of b,c; :: thesis: ( not p1 * f = p2 * f or p1 = p2 )

assume A4: p1 * f = p2 * f ; :: thesis: p1 = p2

thus p1 = p1 * (f * g) by A3, A2, CAT_1:29

.= (p2 * f) * g by A3, A1, A4, CAT_1:25

.= p2 * (f * g) by A3, A1, CAT_1:25

.= p2 by A3, A2, CAT_1:29 ; :: thesis: verum

for f being Morphism of a,b st f is retraction holds

f is epi

let a, b be Object of C; :: thesis: for f being Morphism of a,b st f is retraction holds

f is epi

let f be Morphism of a,b; :: thesis: ( f is retraction implies f is epi )

assume A1: ( Hom (a,b) <> {} & Hom (b,a) <> {} ) ; :: according to CAT_3:def 8 :: thesis: ( for g being Morphism of b,a holds not f * g = id b or f is epi )

given g being Morphism of b,a such that A2: f * g = id b ; :: thesis: f is epi

thus Hom (a,b) <> {} by A1; :: according to CAT_1:def 15 :: thesis: for b

( Hom (b,b

( not b

let c be Object of C; :: thesis: ( Hom (b,c) = {} or for b

( not b

assume A3: Hom (b,c) <> {} ; :: thesis: for b

( not b

let p1, p2 be Morphism of b,c; :: thesis: ( not p1 * f = p2 * f or p1 = p2 )

assume A4: p1 * f = p2 * f ; :: thesis: p1 = p2

thus p1 = p1 * (f * g) by A3, A2, CAT_1:29

.= (p2 * f) * g by A3, A1, A4, CAT_1:25

.= p2 * (f * g) by A3, A1, CAT_1:25

.= p2 by A3, A2, CAT_1:29 ; :: thesis: verum