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 Th23:
  for f,g being FinSequence of TOP-REAL 2 st g is_Shortcut_of f
  holds L~g c= L~f
proof
  let f,g be FinSequence of TOP-REAL 2;
  assume
A1: g is_Shortcut_of f;
  let x be object;
  assume x in L~g;
  then x in union { LSeg(g,i) : 1 <= i & i+1 <= len g } by TOPREAL1:def 4;
  then consider y such that
A2: x in y & y in { LSeg(g,i) : 1 <= i & i+1 <= len g } by TARSKI:def 4;
  consider i such that
A3: y=LSeg(g,i) and
A4: 1<=i & i+1<=len g by A2;
  consider k1 being Element of NAT such that
A5: 1<=k1 & k1+1<=len f and
A6: f/.k1=g/.i & f/.(k1+1)=g/.(i+1) and
  f.k1=g.i and
  f.(k1+1)=g.(i+1) by A1,A4,Th21;
A7: LSeg(f,k1) in {LSeg(f,k):1<=k & k+1<=len f} by A5;
  x in LSeg(g/.i,g/.(i+1)) by A2,A3,A4,TOPREAL1:def 3;
  then x in LSeg(f,k1) by A5,A6,TOPREAL1:def 3;
  then x in union { LSeg(f,k) : 1 <= k & k+1 <= len f } by A7,TARSKI:def 4;
  hence thesis by TOPREAL1:def 4;
end;
