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 Th45:
  for f, g being FinSequence of TOP-REAL 2 st g is_in_the_area_of
  f & i in dom g holds <*g/.i*> is_in_the_area_of f
proof
  let f, g be FinSequence of TOP-REAL 2 such that
A1: g is_in_the_area_of f and
A2: i in dom g;
  let n;
A3: dom <*g/.i*> = {1} by FINSEQ_1:2,38;
  assume n in dom <*g/.i*>;
  then n = 1 by A3,TARSKI:def 1;
  then <*g/.i*>/.n = g/.i by FINSEQ_4:16;
  hence thesis by A1,A2;
end;
