reserve r,s,t,u for Real;

theorem
  for X being RealLinearSpace, M being Subset of X holds M is convex iff
  for u,w being Point of X st u in M & w in M holds LSeg(u,w) c= M
proof
  let X be RealLinearSpace, M be Subset of X;
  hereby
    assume
A1: M is convex;
    let u,w be Point of X such that
A2: u in M & w in M;
    thus LSeg(u,w) c= M
     proof
      let x be object;
      assume x in LSeg(u,w);
      then ex r st x = (1-r)*u + r*w & 0 <= r & r <= 1;
      hence x in M by A1,A2;
     end;
  end;
  assume
A3: for w,u being Point of X st w in M & u in M holds LSeg(w,u) c= M;
  let u,w be Point of X, r be Real such that
A4: 0 <= r & r <= 1 and
A5: u in M & w in M;
A6: r*u + (1-r)*w in LSeg(w,u) by A4;
  LSeg(w,u) c= M by A3,A5;
  hence thesis by A6;
end;
