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;
