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;

theorem Th66:
  for i,j st i in dom f & j in dom f & mid(f,i,j) is
  S-Sequence_in_R2 & f/.i = N-min L~f & N-min L~f <> NW-corner L~f holds <*
  NW-corner L~f*>^mid(f,i,j) is S-Sequence_in_R2
proof
  set p = NW-corner L~f;
  let i,j such that
A1: i in dom f and
A2: j in dom f and
A3: mid(f,i,j) is S-Sequence_in_R2 and
A4: f/.i = N-min L~f and
A5: N-min L~f <> NW-corner L~f;
A6: 1<=i & i<=len f by A1,FINSEQ_3:25;
A7: (mid(f,i,j))/.1 = N-min L~f by A1,A2,A4,Th8;
  then
A8: p`2 = ((mid(f,i,j))/.1)`2 by PSCOMP_1:37;
A9: 1<=j & j<=len f by A2,FINSEQ_3:25;
  len mid(f,i,j) >= 2 by A3,TOPREAL1:def 8;
  then
  LSeg(NW-corner L~f, N-min L~f)/\L~f = {N-min L~f} & N-min L~f in L~mid(f
  ,i,j ) by A7,JORDAN3:1,PSCOMP_1:43;
  then LSeg(p, (mid(f,i,j))/.1) /\ L~mid(f,i,j) = {(mid(f,i,j))/.1} by A7,A6,A9
,JORDAN4:35,ZFMISC_1:124;
  hence thesis by A3,A5,A7,A8,Th60;
end;
