reserve x,y for set;
reserve G for Graph;
reserve vs,vs9 for FinSequence of the carrier of G;
reserve IT for oriented Chain of G;
reserve N for Nat;
reserve n,m,k,i,j for Nat;
reserve r,r1,r2 for Real;
reserve X for non empty set;

theorem Th25:
  for f being FinSequence of TOP-REAL 2 st f is special & 2<=len f
& f.1 <> f.len f holds ex g being FinSequence of TOP-REAL 2 st 2<=len g & g is
  special & g is one-to-one & L~g c= L~f & f.1=g.1 & f.len f=g.len g & rng g c=
  rng f
proof
  let f be FinSequence of TOP-REAL 2;
  assume that
A1: f is special and
A2: 2<=len f and
A3: f.1 <> f.len f;
  consider g being FinSequence of TOP-REAL 2 such that
A4: g is_Shortcut_of f by A2,Th9,XXREAL_0:2;
A5: g.1=f.1 & g.len g=f.len f by A4;
  1<=len g by A4,Th8;
  then 1<len g by A3,A5,XXREAL_0:1;
  then
A6: 1+1<=len g by NAT_1:13;
A7: L~g c= L~f & rng g c= rng f by A4,Th22,Th23;
  g is one-to-one by A3,A4,Th11;
  hence thesis by A1,A4,A7,A6,Th24;
end;
