 reserve n for Nat;

theorem THSS2:
  for a,b,c being Element of OASpace(TOP-REAL 2) holds
    Mid a,b,c iff
      (ex u,v be Point of TOP-REAL 2 st u = a & v = c & b in LSeg(u,v))
  proof
    let a,b,c be Element of OASpace(TOP-REAL 2);
    hereby
      assume Mid a,b,c;
      then per cases by THSS;
      suppose
A1:     a = b;
        reconsider u = a, v = c as Point of TOP-REAL 2;
        u in LSeg(u,v) by RLTOPSP1:68;
        hence ex u,v be Point of TOP-REAL 2 st u = a & v = c &
          b in LSeg(u,v) by A1;
      end;
      suppose
A2:     b = c;
        reconsider u = a, v = c as Point of TOP-REAL 2;
        v in LSeg(u,v) by RLTOPSP1:68;
        hence ex u,v be Point of TOP-REAL 2 st
          u = a & v = c & b in LSeg(u,v) by A2;
      end;
      suppose ex u,v be Point of TOP-REAL 2 st u = a & v = c & b in LSeg(u,v);
        hence ex u,v be Point of TOP-REAL 2 st u = a & v = c & b in LSeg(u,v);
      end;
    end;
    assume ex u,v be Point of TOP-REAL 2 st u = a & v = c & b in LSeg(u,v);
    hence Mid a,b,c by THSS;
  end;
