 reserve n for Nat;

theorem ThConvAGI:
  for a,b,c being Element of TarskiEuclid2Space holds
    (ex jj being Real st 0 <= jj & jj <= 1 &
      Tn2TR b - Tn2TR a = jj * (Tn2TR c - Tn2TR a))
    implies
    Tn2TR b in LSeg (Tn2TR a,Tn2TR c)
  proof
    let a,b,c be Element of TarskiEuclid2Space;
    given jj being Real such that
G2: 0 <= jj <= 1 & Tn2TR b - Tn2TR a = jj * (Tn2TR c - Tn2TR a);
    set v = Tn2TR a, w = Tn2TR c;
SS: Tn2TR b + (v - v) = Tn2TR b + 0.TOP-REAL 2 by RLVECT_1:5
                .= Tn2TR b;
G3: v = 1 * v by RVSUM_1:52;
    Tn2TR b + (- v + v) = jj * (Tn2TR c - Tn2TR a) + v by G2,RVSUM_1:15; then
    Tn2TR b = jj*w + jj * (-v) + v by RVSUM_1:51,SS
      .= jj*w + (jj * (-1))*v + v by RVSUM_1:49
      .= jj*w + (-1) * (jj*v) + v by RVSUM_1:49
      .= jj*w + (- jj*v + 1*v) by RVSUM_1:15,G3
      .= jj*w + (((-1) * jj)*v + 1*v) by RVSUM_1:49
      .= ((1 - jj)*v) + jj*w by RVSUM_1:50; then
    Tn2TR b in {(1-r)*v + r*w where r is Real : 0 <= r & r <= 1 } by G2;
    hence thesis by RLTOPSP1:def 2;
  end;
