reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem Th8:
  for f being FinSequence of REAL, A being finite Subset of REAL
  st A = rng f holds SgmX(RealOrd,A) = Incr f
proof
  let f be FinSequence of REAL, A be finite Subset of REAL such that
A1: A = rng f;
  reconsider F = SgmX(RealOrd,A) as increasing FinSequence of REAL by Th7;
  RealOrd linearly_orders A by Th6,ORDERS_1:38;
  then
A2: rng SgmX(RealOrd,A) = rng f by A1,PRE_POLY:def 2;
  len F = card rng f by A1,Th6,ORDERS_1:38,PRE_POLY:11;
  hence thesis by A2,SEQ_4:def 21;
end;
