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
  for f being Permutation of the carrier of OAS st f is dilatation ex f9
being Permutation of the carrier of Lambda(OAS) st f=f9 & f9 is_DIL_of Lambda(
  OAS)
proof
  let f be Permutation of the carrier of OAS;
A1: Lambda(OAS) = AffinStruct(#the carrier of OAS, lambda(the CONGR of OAS)
  #) by DIRAF:def 2;
  then reconsider f9=f as Permutation of the carrier of Lambda(OAS);
  assume f is dilatation;
  then
A2: f is_FormalIz_of lambda(the CONGR of OAS);
  take f9;
  thus thesis by A2,A1;
end;
