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;
