reserve p,p1,p2,q for Point of TOP-REAL 2,
  f,f1,f2,g,g1,g2 for FinSequence of TOP-REAL 2,
  r,s for Real,

  n,m,i,j,k for Nat,
  G for Go-board,
  x for set;

theorem Th6:
  L~f c= L~(f^g)
proof
  set f1 = f^g, lf = {LSeg(f,i): 1<=i & i+1 <= len f}, l1 = {LSeg(f1,j): 1<=j
  & j+1 <= len f1};
  let x be object;
  assume x in L~f;
  then consider X be set such that
A1: x in X and
A2: X in lf by TARSKI:def 4;
  consider n such that
A3: X=LSeg(f,n) and
A4: 1<=n and
A5: n+1 <= len f by A2;
  n<=n+1 by NAT_1:11;
  then n<=len f by A5,XXREAL_0:2;
  then
A6: n in dom f by A4,FINSEQ_3:25;
  len f1=len f +len g by FINSEQ_1:22;
  then len f <= len f1 by XREAL_1:31;
  then
A7: n+1 <= len f1 by A5,XXREAL_0:2;
  1<=n+1 by XREAL_1:31;
  then n+1 in dom f by A5,FINSEQ_3:25;
  then X=LSeg(f1,n) by A3,A6,TOPREAL3:18;
  then X in l1 by A4,A7;
  hence thesis by A1,TARSKI:def 4;
end;
