let C be non empty transitive associative AltCatStr ; :: thesis: for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is epi & B is epi holds
B * A is epi
let o1, o2, o3 be object of C; :: thesis: ( <^o1,o2^> <> {} & <^o2,o3^> <> {} implies for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is epi & B is epi holds
B * A is epi )
assume A1: ( <^o1,o2^> <> {} & <^o2,o3^> <> {} ) ; :: thesis: for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is epi & B is epi holds
B * A is epi
let A be Morphism of o1,o2; :: thesis: for B being Morphism of o2,o3 st A is epi & B is epi holds
B * A is epi
let B be Morphism of o2,o3; :: thesis: ( A is epi & B is epi implies B * A is epi )
assume A2: ( A is epi & B is epi ) ; :: thesis: B * A is epi
let o be object of C; :: according to ALTCAT_3:def 8 :: thesis: ( <^o3,o^> <> {} implies for B, C being Morphism of o3,o st B * (B * A) = C * (B * A) holds
B = C )
assume A3: <^o3,o^> <> {} ; :: thesis: for B, C being Morphism of o3,o st B * (B * A) = C * (B * A) holds
B = C
let M1, M2 be Morphism of o3,o; :: thesis: ( M1 * (B * A) = M2 * (B * A) implies M1 = M2 )
assume A4: M1 * (B * A) = M2 * (B * A) ; :: thesis: M1 = M2
A5: M1 * (B * A) = (M1 * B) * A by A1, A3, ALTCAT_1:25;
A6: M2 * (B * A) = (M2 * B) * A by A1, A3, ALTCAT_1:25;
<^o2,o^> <> {} by A1, A3, ALTCAT_1:def 4;
then M1 * B = M2 * B by A2, A4, A5, A6, Def8;
hence M1 = M2 by A2, A3, Def8; :: thesis: verum