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;

theorem Th37:
  for f being non constant standard special_circular_sequence, i,j
  st i > j & (1 < j & i <= len f or 1 <= j & i < len f) holds mid(f,i,j) is
  S-Sequence_in_R2
proof
  let f be non constant standard special_circular_sequence, i,j such that
A1: i > j and
A2: 1 < j & i <= len f or 1 <= j & i < len f;
A3: Rev mid(f,j,i) = mid(f,i,j) by FINSEQ_6:196;
  per cases by A2;
  suppose
    1 < j & i <= len f;
    then mid(f,j,i) is S-Sequence_in_R2 by A1,JORDAN4:40;
    hence thesis by A3;
  end;
  suppose
A4: 1 <= j & i < len f;
    then i+1 <= len f by NAT_1:13;
    then mid(f,j,i) is S-Sequence_in_R2 by A1,A4,JORDAN4:39;
    hence thesis by A3;
  end;
end;
