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
  f is positive_dilatation or f is negative_dilatation implies f is dilatation
proof
  assume
A1: f is positive_dilatation or f is negative_dilatation;
  now
    let x,y;
    x,y // f.x,f.y or x,y // f.y,f.x by A1,Th27;
    hence x,y '||' f.x,f.y by DIRAF:def 4;
  end;
  hence thesis by Th34;
end;
