let AS be non empty AffinStruct ; :: thesis: for f being Permutation of the carrier of AS holds
( f is_DIL_of AS iff for a, b being Element of AS holds a,b // f . a,f . b )
let f be Permutation of the carrier of AS; :: thesis: ( f is_DIL_of AS iff for a, b being Element of AS holds a,b // f . a,f . b )
A1: now :: thesis: ( ( for a, b being Element of AS holds a,b // f . a,f . b ) implies f is_DIL_of AS )
assume A2: for a, b being Element of AS holds a,b // f . a,f . b ; :: thesis: f is_DIL_of AS
for x, y being Element of AS holds [[x,y],[(f . x),(f . y)]] in the CONGR of AS by A2, ANALOAF:def 2;
then f is_FormalIz_of the CONGR of AS ;
hence f is_DIL_of AS ; :: thesis: verum
end;
now :: thesis: ( f is_DIL_of AS implies for a, b being Element of AS holds a,b // f . a,f . b )
assume A3: f is_DIL_of AS ; :: thesis: for a, b being Element of AS holds a,b // f . a,f . b
let a, b be Element of AS; :: thesis: a,b // f . a,f . b
f is_FormalIz_of the CONGR of AS by A3;
then [[a,b],[(f . a),(f . b)]] in the CONGR of AS ;
hence a,b // f . a,f . b by ANALOAF:def 2; :: thesis: verum
end;
hence ( f is_DIL_of AS iff for a, b being Element of AS holds a,b // f . a,f . b ) by A1; :: thesis: verum