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;
reserve p,p1,p2 for Point of TOP-REAL N;
reserve M for non empty MetrSpace;

theorem Th40:
  for f being FinSequence of TOP-REAL 2 st 1 <=len f holds len
  X_axis(f)=len f & X_axis(f).1=(f/.1)`1 & X_axis(f).len f=(f/.len f)`1
proof
  let f be FinSequence of TOP-REAL 2;
A1: len (X_axis(f)) = len f by GOBOARD1:def 1;
  assume
A2: 1 <=len f;
  then len f in Seg len f by FINSEQ_1:1;
  then
A3: len f in dom (X_axis(f)) by A1,FINSEQ_1:def 3;
  1 in Seg len f by A2,FINSEQ_1:1;
  then 1 in dom (X_axis(f)) by A1,FINSEQ_1:def 3;
  hence thesis by A3,GOBOARD1:def 1;
end;
