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 Th68:
  f/.1 = N-min L~f implies (N-min L~f)..f < (N-max L~f)..f
proof
  assume f/.1 = N-min L~f;
  then
A1: (N-min L~f)..f = 1 by FINSEQ_6:43;
A2: N-max L~f in rng f by Th40;
  then (N-max L~f)..f in dom f by FINSEQ_4:20;
  then
A3: (N-max L~f)..f >= 1 by FINSEQ_3:25;
  N-min L~f in rng f by Th39;
  then (N-min L~f)..f <> (N-max L~f)..f by A2,Th52,FINSEQ_5:9;
  hence thesis by A3,A1,XXREAL_0:1;
end;
