reserve i,j,k,n for Nat,
  X,Y,a,b,c,x for set,
  r,s for Real;
reserve f,g for FinSequence of TOP-REAL 2;

theorem Th8:
  for k,f,g st 1<=k & k+1<=len g & f,g are_in_general_position
  holds g.k in (L~f)` & g.(k+1) in (L~f)`
proof
  let k,f,g such that
A1: 1<=k and
A2: k+1<=len g and
A3: f,g are_in_general_position;
  f is_in_general_position_wrt g by A3;
  then
A4: L~f misses rng g;
A5: rng g c= the carrier of TOP-REAL 2 by FINSEQ_1:def 4;
  k < len g by A2,NAT_1:13;
  then k in dom g by A1,FINSEQ_3:25;
  then
A6: g.k in rng g by FUNCT_1:3;
  now
    assume not g.k in (L~f)`;
    then g.k in (L~f)`` by A6,A5,XBOOLE_0:def 5;
    hence contradiction by A4,A6,XBOOLE_0:3;
  end;
  hence g.k in (L~f)`;
  1<=k+1 by A1,NAT_1:13;
  then k+1 in dom g by A2,FINSEQ_3:25;
  then
A7: g.(k+1) in rng g by FUNCT_1:3;
  now
    assume not g.(k+1) in (L~f)`;
    then g.(k+1) in (L~f)`` by A5,A7,XBOOLE_0:def 5;
    hence contradiction by A4,A7,XBOOLE_0:3;
  end;
  hence thesis;
end;
