 reserve n for Nat;

theorem THSS3:
  for a,b,c being Element of OASpace(TOP-REAL 2),
      ap,bp,cp being POINT of TarskiEuclid2Space st
        a = ap & b = bp & c = cp holds Mid a,b,c iff between ap,bp,cp
  proof
    let a,b,c be Element of OASpace(TOP-REAL 2),
        ap,bp,cp be POINT of TarskiEuclid2Space;
    assume that
A1: a = ap and
A2: b = bp and
A3: c = cp;
    hereby
      assume Mid a,b,c;
      then ex u,v be Point of TOP-REAL 2 st u = a & v = c & b in LSeg(u,v)
        by THSS2;
      then Tn2TR bp in LSeg(Tn2TR ap, Tn2TR cp) by A1,A2,A3;
      hence between ap,bp,cp by ThConv6;
    end;
    assume between ap,bp,cp;
    then Tn2TR bp in LSeg(Tn2TR ap, Tn2TR cp) by ThConv6;
    hence Mid a,b,c by A1,A2,A3,THSS2;
  end;
