let C be category; for a, b, c being Object of C
for f1 being Morphism of a,b
for f2 being Morphism of b,c st f1 is monomorphism & f2 is monomorphism holds
f2 * f1 is monomorphism
let a, b, c be Object of C; for f1 being Morphism of a,b
for f2 being Morphism of b,c st f1 is monomorphism & f2 is monomorphism holds
f2 * f1 is monomorphism
let f1 be Morphism of a,b; for f2 being Morphism of b,c st f1 is monomorphism & f2 is monomorphism holds
f2 * f1 is monomorphism
let f2 be Morphism of b,c; ( f1 is monomorphism & f2 is monomorphism implies f2 * f1 is monomorphism )
assume A1:
f1 is monomorphism
; ( not f2 is monomorphism or f2 * f1 is monomorphism )
assume A2:
f2 is monomorphism
; f2 * f1 is monomorphism
hence
Hom (a,c) <> {}
by A1, Th22; CAT_7:def 5 for c being Object of C st Hom (c,a) <> {} holds
for g1, g2 being Morphism of c,a st (f2 * f1) * g1 = (f2 * f1) * g2 holds
g1 = g2
let d be Object of C; ( Hom (d,a) <> {} implies for g1, g2 being Morphism of d,a st (f2 * f1) * g1 = (f2 * f1) * g2 holds
g1 = g2 )
assume A3:
Hom (d,a) <> {}
; for g1, g2 being Morphism of d,a st (f2 * f1) * g1 = (f2 * f1) * g2 holds
g1 = g2
let g1, g2 be Morphism of d,a; ( (f2 * f1) * g1 = (f2 * f1) * g2 implies g1 = g2 )
assume A4:
(f2 * f1) * g1 = (f2 * f1) * g2
; g1 = g2
A5:
Hom (d,b) <> {}
by A3, A1, Th22;
f2 * (f1 * g1) =
(f2 * f1) * g1
by A1, A2, A3, Th23
.=
f2 * (f1 * g2)
by A4, A1, A2, A3, Th23
;
hence
g1 = g2
by A1, A3, A2, A5; verum