let A, B be category; :: thesis: ( A,B are_opposite implies for a, b being object of A st <^a,b^> <> {} & <^b,a^> <> {} holds
for a', b' being object of B st a' = a & b' = b holds
for f being Morphism of a,b
for f' being Morphism of b',a' st f' = f & f is retraction & f is coretraction holds
f' " = f " )
assume A1: A,B are_opposite ; :: thesis: for a, b being object of A st <^a,b^> <> {} & <^b,a^> <> {} holds
for a', b' being object of B st a' = a & b' = b holds
for f being Morphism of a,b
for f' being Morphism of b',a' st f' = f & f is retraction & f is coretraction holds
f' " = f "
let a, b be object of A; :: thesis: ( <^a,b^> <> {} & <^b,a^> <> {} implies for a', b' being object of B st a' = a & b' = b holds
for f being Morphism of a,b
for f' being Morphism of b',a' st f' = f & f is retraction & f is coretraction holds
f' " = f " )
assume A2: ( <^a,b^> <> {} & <^b,a^> <> {} ) ; :: thesis: for a', b' being object of B st a' = a & b' = b holds
for f being Morphism of a,b
for f' being Morphism of b',a' st f' = f & f is retraction & f is coretraction holds
f' " = f "
let a', b' be object of B; :: thesis: ( a' = a & b' = b implies for f being Morphism of a,b
for f' being Morphism of b',a' st f' = f & f is retraction & f is coretraction holds
f' " = f " )
assume A3: ( a' = a & b' = b ) ; :: thesis: for f being Morphism of a,b
for f' being Morphism of b',a' st f' = f & f is retraction & f is coretraction holds
f' " = f "
A4: ( <^b',a'^> = <^a,b^> & <^a',b'^> = <^b,a^> ) by A1, A3, Th9;
let f be Morphism of a,b; :: thesis: for f' being Morphism of b',a' st f' = f & f is retraction & f is coretraction holds
f' " = f "
let f' be Morphism of b',a'; :: thesis: ( f' = f & f is retraction & f is coretraction implies f' " = f " )
assume A5: ( f' = f & f is retraction & f is coretraction ) ; :: thesis: f' " = f "
reconsider g = f " as Morphism of a',b' by A1, A3, Th7;
( (f " ) * f = idm a & f * (f " ) = idm b ) by A2, A5, ALTCAT_3:2;
then ( f' * g = idm a & g * f' = idm b ) by A1, A2, A3, A5, Th9;
then ( f' * g = idm a' & g * f' = idm b' ) by A1, A3, Th10;
then ( f' is retraction & f' is coretraction & g is_left_inverse_of f' & g is_right_inverse_of f' ) by A1, A2, A3, A5, Lm1, ALTCAT_3:def 1;
hence f' " = f " by A2, A4, ALTCAT_3:def 4; :: thesis: verum