let A, B be category; :: thesis: ( A,B are_opposite implies for a, b being object of st <^a,b^> <> {} & <^b,a^> <> {} holds
for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f holds
( f is iso iff f' is iso ) )
assume A1: A,B are_opposite ; :: thesis: for a, b being object of st <^a,b^> <> {} & <^b,a^> <> {} holds
for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f holds
( f is iso iff f' is iso )
let a, b be object of ; :: thesis: ( <^a,b^> <> {} & <^b,a^> <> {} implies for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f holds
( f is iso iff f' is iso ) )
assume that
A2: <^a,b^> <> {} and
A3: <^b,a^> <> {} ; :: thesis: for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f holds
( f is iso iff f' is iso )
let a', b' be object of ; :: thesis: ( a' = a & b' = b implies for f being Morphism of ,
for f' being Morphism of , st f' = f holds
( f is iso iff f' is iso ) )
assume that
A4: a' = a and
A5: b' = b ; :: thesis: for f being Morphism of ,
for f' being Morphism of , st f' = f holds
( f is iso iff f' is iso )
A6: <^b',a'^> = <^a,b^> by A1, A4, A5, Th9;
A7: <^a',b'^> = <^b,a^> by A1, A4, A5, Th9;

now

let A, B be category; :: thesis: ( A,B are_opposite implies for a, b being object of st <^a,b^> <> {} & <^b,a^> <> {} holds
for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f & f is iso holds
f' is iso )
assume A8: A,B are_opposite ; :: thesis: for a, b being object of st <^a,b^> <> {} & <^b,a^> <> {} holds
for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f & f is iso holds
f' is iso
let a, b be object of ; :: thesis: ( <^a,b^> <> {} & <^b,a^> <> {} implies for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f & f is iso holds
f' is iso )
assume that
A9: <^a,b^> <> {} and
A10: <^b,a^> <> {} ; :: thesis: for a', b' being object of st a' = a & b' = b holds
for f being Morphism of ,
for f' being Morphism of , st f' = f & f is iso holds
f' is iso
let a', b' be object of ; :: thesis: ( a' = a & b' = b implies for f being Morphism of ,
for f' being Morphism of , st f' = f & f is iso holds
f' is iso )
assume that
A11: a' = a and
A12: b' = b ; :: thesis: for f being Morphism of ,
for f' being Morphism of , st f' = f & f is iso holds
f' is iso
let f be Morphism of ,; :: thesis: for f' being Morphism of , st f' = f & f is iso holds
f' is iso
let f' be Morphism of ,; :: thesis: ( f' = f & f is iso implies f' is iso )
assume A13: f' = f ; :: thesis: ( f is iso implies f' is iso )
assume A14: f is iso ; :: thesis: f' is iso
then A15: f * (f " ) = idm b by ALTCAT_3:def 5;
A16: (f " ) * f = idm a by A14, ALTCAT_3:def 5;
( f is retraction & f is coretraction ) by A14, ALTCAT_3:5;
then A17: f' " = f " by A8, A9, A10, A11, A12, A13, Th23;
A18: idm a = idm a' by A8, A11, Th10;
A19: idm b = idm b' by A8, A12, Th10;
A20: f' * (f' " ) = idm a' by A8, A9, A10, A11, A12, A13, A16, A17, A18, Th9;
(f' " ) * f' = idm b' by A8, A9, A10, A11, A12, A13, A15, A17, A19, Th9;
hence f' is iso by A20, ALTCAT_3:def 5; :: thesis: verum

end;

hence for f being Morphism of ,
for f' being Morphism of , st f' = f holds
( f is iso iff f' is iso ) by A1, A2, A3, A4, A5, A6, A7; :: thesis: verum