 reserve n for Nat;

theorem ThConvAG:
  for a,b,c being Element of TarskiEuclid2Space holds
    Tn2TR b in LSeg (Tn2TR a,Tn2TR c) implies
    ex jj being Real st 0 <= jj & jj <= 1 &
      Tn2TR b - Tn2TR a = jj * (Tn2TR c - Tn2TR a)
  proof
    let a,b,c be Element of TarskiEuclid2Space;
    assume Tn2TR b in LSeg (Tn2TR a,Tn2TR c); then
    consider jj being Real such that
G2: 0 <= jj & jj <= 1 & Tn2TR b = (1-jj)*Tn2TR a + jj*Tn2TR c by RLTOPSP1:76;
    set v = Tn2TR a, w = Tn2TR c;
    Tn2TR b - Tn2TR a = (1-jj)*v - v + jj*w by RVSUM_1:15,G2
           .= (1-jj+(-1))*v + jj*w by RVSUM_1:50
           .= jj*w + (jj*(-1))*v
           .= jj*w + jj*((-1)*v) by RVSUM_1:49
           .= jj * (w - v) by RVSUM_1:51;
    hence thesis by G2;
  end;
