reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem
  for f being S-Sequence_in_R2 st p in rng f & p..f <> len f holds f:-p
  is being_S-Seq
proof
  let f be S-Sequence_in_R2 such that
A1: p in rng f and
A2: p..f <> len f;
  thus f:-p is one-to-one by A1,FINSEQ_5:56;
  hereby
    p..f <= len f by A1,FINSEQ_4:21;
    then p..f < len f by A2,XXREAL_0:1;
    then 1 + p..f <= len f by NAT_1:13;
    then
A3: len f - p..f >= 1 by XREAL_1:19;
    assume len(f:-p) < 2;
    then len f - p..f + 1 < 1 + 1 by A1,FINSEQ_5:50;
    hence contradiction by A3,XREAL_1:6;
  end;
  thus thesis by A1,Th27,Th34,Th39;
end;
