reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem Th11:
  for f being FinSequence of TOP-REAL 2 holds Y_axis f = proj2*f
proof
  let f be FinSequence of TOP-REAL 2;
  reconsider pf = proj2*f as FinSequence of REAL by FINSEQ_2:32;
A1: len Y_axis f = len f by GOBOARD1:def 2;
  then
A2: dom Y_axis f = dom f by FINSEQ_3:29;
A3: for k being Nat st k in dom Y_axis f holds (Y_axis f).k = pf.k
  proof
    let k be Nat such that
A4: k in dom Y_axis f;
A5: f/.k = f.k by A2,A4,PARTFUN1:def 6;
    thus (Y_axis f).k = (f/.k)`2 by A4,GOBOARD1:def 2
      .= proj2.(f.k) by A5,PSCOMP_1:def 6
      .= pf.k by A2,A4,FUNCT_1:13;
  end;
  len pf = len f by FINSEQ_2:33;
  then dom Y_axis f = dom pf by A1,FINSEQ_3:29;
  hence thesis by A3,FINSEQ_1:13;
end;
