reserve p,p1,p2,q for Point of TOP-REAL 2,
  f,g,g1,g2 for FinSequence of TOP-REAL 2,
  n,m,i,j,k for Nat,
  G for Go-board,
  x for set;

theorem
  (for n st n in dom f ex i,j st [i,j] in Indices G & f/.n=G*(i,j)) & f
is being_S-Seq implies ex g st g is_sequence_on G & g is being_S-Seq & L~f = L~
  g & f/.1 = g/.1 & f/.len f = g/.len g & len f<=len g
proof
  assume that
A1: for n st n in dom f ex i,j st [i,j] in Indices G & f/.n=G*(i,j) and
A2: f is being_S-Seq;
  f is one-to-one & f is unfolded s.n.c. special by A2;
  then consider g such that
A3: g is_sequence_on G and
A4: g is one-to-one unfolded s.n.c. special and
A5: L~f = L~g & f/.1 = g/.1 & f/.len f = g/.len g and
A6: len f <= len g by A1,Th1;
  take g;
  thus g is_sequence_on G by A3;
  2<=len f by A2;
  then 2<=len g by A6,XXREAL_0:2;
  hence g is being_S-Seq by A4;
  thus thesis by A5,A6;
end;
