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;
reserve CS for CongrSpace;
reserve OAS for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u for Element of OAS;
reserve f,g for Permutation of the carrier of OAS;

theorem Th34:
  for f being Permutation of the carrier of OAS holds (f is
  dilatation iff for a,b holds a,b '||' f.a,f.b )
proof
  let f be Permutation of the carrier of OAS;
A1: now
    assume
A2: for a,b holds a,b '||' f.a,f.b;
    for a,b holds [[a,b],[f.a,f.b]] in lambda (the CONGR of OAS)
      by A2,DIRAF:18;
    then f is_FormalIz_of lambda(the CONGR of OAS);
    hence f is dilatation;
  end;
  now
    assume
A3: f is dilatation;
    let a,b;
    f is_FormalIz_of lambda(the CONGR of OAS) by A3;
    then [[a,b],[f.a,f.b]] in lambda(the CONGR of OAS);
    hence a,b '||' f.a,f.b by DIRAF:18;
  end;
  hence thesis by A1;
end;
