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
 for x1,x2,x3,x4 be Point of V
  st x1 <> x2 & x1,x2,x3 are_collinear & x1,x2,x4 are_collinear
 holds x1,x2,x3,x4 are_collinear
proof let x1,x2,x3,x4 be Point of V such that
A1: x1 <> x2;
 given L1 being line of V such that
A2: x1 in L1 & x2 in L1 & x3 in L1;
 given L2 being line of V such that
A3: x1 in L2 & x2 in L2 & x4 in L2;
  L1 = Line(x1,x2) & L2 = Line(x1,x2) by A1,A2,A3,Th80;
 hence ex L being line of V st x1 in L & x2 in L & x3 in L & x4 in L
     by A2,A3;
end;
