let C be category; :: thesis: for o1, o2, o3 being Object of C

for v being Morphism of o2,o3

for u being Morphism of o1,o3

for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let o1, o2, o3 be Object of C; :: thesis: for v being Morphism of o2,o3

for u being Morphism of o1,o3

for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let v be Morphism of o2,o3; :: thesis: for u being Morphism of o1,o3

for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let u be Morphism of o1,o3; :: thesis: for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let f be Morphism of o1,o2; :: thesis: ( u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} implies v = u * (f ") )

assume that

A1: u = v * f and

A2: f * (f ") = idm o2 and

A3: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} ) and

A4: <^o2,o3^> <> {} ; :: thesis: v = u * (f ")

thus u * (f ") = v * (f * (f ")) by A1, A3, A4, ALTCAT_1:21

.= v by A2, A4, ALTCAT_1:def 17 ; :: thesis: verum

for v being Morphism of o2,o3

for u being Morphism of o1,o3

for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let o1, o2, o3 be Object of C; :: thesis: for v being Morphism of o2,o3

for u being Morphism of o1,o3

for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let v be Morphism of o2,o3; :: thesis: for u being Morphism of o1,o3

for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let u be Morphism of o1,o3; :: thesis: for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds

v = u * (f ")

let f be Morphism of o1,o2; :: thesis: ( u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} implies v = u * (f ") )

assume that

A1: u = v * f and

A2: f * (f ") = idm o2 and

A3: ( <^o1,o2^> <> {} & <^o2,o1^> <> {} ) and

A4: <^o2,o3^> <> {} ; :: thesis: v = u * (f ")

thus u * (f ") = v * (f * (f ")) by A1, A3, A4, ALTCAT_1:21

.= v by A2, A4, ALTCAT_1:def 17 ; :: thesis: verum