theorem Th41:
  for f being FinSequence of TOP-REAL 2 st 1 <=len f holds len
  Y_axis(f)=len f & Y_axis(f).1=(f/.1)`2 & Y_axis(f).len f=(f/.len f)`2
proof
  let f be FinSequence of TOP-REAL 2;
A1: len (Y_axis(f)) = len f by GOBOARD1:def 2;
  assume
A2: 1 <=len f;
  then len f in Seg len f by FINSEQ_1:1;
  then
A3: len f in dom (Y_axis(f)) by A1,FINSEQ_1:def 3;
  1 in Seg len f by A2,FINSEQ_1:1;
  then 1 in dom (Y_axis(f)) by A1,FINSEQ_1:def 3;
  hence thesis by A3,GOBOARD1:def 2;
end;
