reserve r,s,t,u for Real;
reserve V for RealLinearSpace,
  v,w for Point of V;
reserve x1,x2,x3,x4,y1,y2 for Element of V;

theorem Th77:
 Line(v,v) = {v}
 proof
  for x being object holds x in Line(v,v) iff x = v
   proof let x be object;
    thus x in Line(v,v) implies x = v
     proof assume
A1:     x in Line(v,v);
      then reconsider w=x as Point of V;
      consider r such that
A2:     w = (1-r)*v + r*v by A1;
       w = ((1-r)+r)*v by A2,RLVECT_1:def 6;
      hence x = v by RLVECT_1:def 8;
     end;
    assume x = v;
    hence thesis by Th72;
   end;
  hence thesis by TARSKI:def 1;
 end;
