reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem Th7:
  for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
  2, i being Element of NAT st p in LSeg(f,i) holds Index(p,f) <= i
proof
  let f being FinSequence of TOP-REAL 2;
  let p being Point of TOP-REAL 2, i0 be Element of NAT;
  assume
A1: p in LSeg(f,i0);
  LSeg(f,i0) c= L~f by TOPREAL3:19;
  then consider S being non empty Subset of NAT such that
A2: Index(p,f) = min S and
A3: S = { i: p in LSeg(f,i) } by A1,Def1;
  i0 in S by A1,A3;
  hence thesis by A2,XXREAL_2:def 7;
end;
