reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem Th46:
  for f, g being FinSequence of TOP-REAL 2, p being Point of
TOP-REAL 2 st g is_in_the_area_of f & p in L~g holds <*p*> is_in_the_area_of f
proof
  let f, g be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2 such that
A1: g is_in_the_area_of f;
  assume p in L~g;
  then consider i such that
A2: 1 <= i and
A3: i+1 <= len g and
A4: p in LSeg(g/.i,g/.(i+1)) by SPPOL_2:14;
A5: ex r being Real st p = (1-r)*g/.i+r*g/.(i+1) & 0<=r & r<=1 by A4;
  i <= i+1 by NAT_1:11;
  then i <= len g by A3,XXREAL_0:2;
  then i in dom g by A2,FINSEQ_3:25;
  then
A6: <*g/.i*> is_in_the_area_of f by A1,Th45;
  1 <= i+1 by NAT_1:11;
  then i+1 in dom g by A3,FINSEQ_3:25;
  then <*g/.(i+1)*> is_in_the_area_of f by A1,Th45;
  then <*g/.i,g/.(i+1)*> is_in_the_area_of f by A6,Th42;
  hence thesis by A5,Th44;
end;
