theorem Th63:
  f is dilatation & f.p=p & q<>p & Mid q,p,f.q implies f is negative_dilatation
proof
  assume
A1: f is dilatation & f.p=p & q<>p & Mid q,p,f.q;
  x,y // f.y,f.x
  proof
    not p,x,y are_collinear implies x,y // f.y,f.x by A1,Th61;
    hence thesis by A1,Th62;
  end;
  hence thesis;
end;
