reserve i,j,k,l,m,n for Nat,
  D for non empty set,
  f for FinSequence of D;
reserve X for compact Subset of TOP-REAL 2;
reserve r for Real;
reserve f for non trivial FinSequence of TOP-REAL 2;
reserve f for non constant standard special_circular_sequence;
reserve z for clockwise_oriented non constant standard
  special_circular_sequence;

theorem
  f/.1 = N-min L~f implies (W-max L~f)..f < len f
proof
  assume
A1: f/.1 = N-min L~f;
  then
A2: f/.len f = N-min L~f by FINSEQ_6:def 1;
A3: W-max L~f in rng f by Th44;
  then (W-max L~f)..f in dom f by FINSEQ_4:20;
  then
A4: f/.((W-max L~f)..f) = f.((W-max L~f)..f) by PARTFUN1:def 6
    .= W-max L~f by A3,FINSEQ_4:19;
  per cases;
  suppose
    N-min L~f = W-max L~f;
    then (W-max L~f)..f = 1 by A1,FINSEQ_6:43;
    hence thesis by GOBOARD7:34,XXREAL_0:2;
  end;
  suppose
A5: N-min L~f <> W-max L~f;
    (W-max L~f)..f <= len f by A3,FINSEQ_4:21;
    hence thesis by A2,A4,A5,XXREAL_0:1;
  end;
end;
