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 Th42:
  for f being FinSequence of TOP-REAL 2 st i in dom f holds X_axis
  (f).i=(f/.i)`1 & Y_axis(f).i=(f/.i)`2
proof
  let f be FinSequence of TOP-REAL 2;
  assume
A1: i in dom f;
  len (X_axis(f)) = len f by GOBOARD1:def 1;
  then i in dom (X_axis(f)) by A1,FINSEQ_3:29;
  hence (X_axis(f)).i = (f/.i)`1 by GOBOARD1:def 1;
  len (Y_axis(f)) = len f by GOBOARD1:def 2;
  then i in dom Y_axis f by A1,FINSEQ_3:29;
  hence thesis by GOBOARD1:def 2;
end;
