reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];
reserve AS for non empty AffinStruct;
reserve a,b,x,y for Element of AS;

theorem Th22:
  for f being Permutation of the carrier of AS holds (f is_DIL_of
  AS iff for a,b holds a,b // f.a,f.b )
proof
  let f be Permutation of the carrier of AS;
A1: now
    assume
A2: for a,b holds a,b // f.a,f.b;
    for x,y 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;
  end;
  now
    assume
A3: f is_DIL_of AS;
    let a,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;
  end;
  hence thesis by A1;
end;
