reserve n for Nat;

theorem Th1:
  for f,g be FinSequence of TOP-REAL 2 st f is_in_the_area_of g for
  p be Element of TOP-REAL 2 st p in rng f holds f-:p is_in_the_area_of g
proof
  let f,g be FinSequence of TOP-REAL 2;
  assume
A1: f is_in_the_area_of g;
  let p be Element of TOP-REAL 2;
  assume
A2: p in rng f;
  then
A3: p..f <= len f by FINSEQ_4:21;
  let n be Nat;
  assume
A4: n in dom(f-:p);
A5: len(f-:p) = p..f by A2,FINSEQ_5:42;
  then n in Seg(p..f) by A4,FINSEQ_1:def 3;
  then
A6: (f-:p)/.n = f/.n by A2,FINSEQ_5:43;
A7: n in Seg len(f-:p) by A4,FINSEQ_1:def 3;
  then n <= p..f by A5,FINSEQ_1:1;
  then
A8: n <= len f by A3,XXREAL_0:2;
  1 <= n by A7,FINSEQ_1:1;
  then n in dom f by A8,FINSEQ_3:25;
  hence thesis by A1,A6;
end;
