reserve S for Subset of TOP-REAL 2,
  C,C1,C2 for non empty compact Subset of TOP-REAL 2,
  p,q for Point of TOP-REAL 2;

theorem Th8:
  for p,q,r being Point of TOP-REAL 2 holds L~<*p,q,r*> = LSeg(p,q
  ) \/ LSeg(q,r)
proof
  let p,q,r be Point of TOP-REAL 2;
A1: <*p,q*>/.2 = q by FINSEQ_4:17;
A2: <*r*>/.1 = r by FINSEQ_4:16;
A3: <*p,q,r*> = <*p,q*>^<*r*> by FINSEQ_1:43;
  len <*p,q*> = 2 by FINSEQ_1:44;
  hence L~<*p,q,r*> = L~<*p,q*> \/ LSeg(q,r) \/ L~<*r*> by A1,A2,A3,SPPOL_2:23
    .= L~<*p,q*> \/ LSeg(q,r) \/ {} by SPPOL_2:12
    .= LSeg(p,q) \/ LSeg(q,r) by SPPOL_2:21;
end;
