theorem
  P is_strongly_connected_in X iff P is_reflexive_in X & P is_connected_in X
proof
  thus P is_strongly_connected_in X implies P is_reflexive_in X & P
  is_connected_in X;
  assume that
A1: P is_reflexive_in X and
A2: P is_connected_in X;
  let x,y be object;
  assume that
A3: x in X and
A4: y in X;
  x = y implies thesis by A1,A3;
  hence thesis by A2,A3,A4;
end;
