reserve FT for non empty RelStr,
  A,B,C for Subset of FT;

theorem Th46:
  for g being FinSequence of FT, k being Nat st g is continuous &
  1<=k holds g|k is continuous
proof
  let g be FinSequence of FT, k be Nat;
  assume that
A1: g is continuous and
A2: 1<=k;
  per cases;
  suppose
    len g <= k;
    hence thesis by A1,FINSEQ_1:58;
  end;
  suppose
A3: k<=len g;
    hence len(g|k) >= 1 by A2,FINSEQ_1:59;
    let i be Nat,x11 be Element of FT;
    assume that
A4: 1<=i and
A5: i<len (g|k) and
A6: x11=(g|k).i;
A7: len (g|k)=k by A3,FINSEQ_1:59;
    then
A8: (g|k).i=g.i by A5,FINSEQ_3:112;
    i+1<=k by A7,A5,NAT_1:13;
    then
A9: (g|k).(i+1)=g.(i+1) by FINSEQ_3:112;
    i<len g by A3,A7,A5,XXREAL_0:2;
    hence thesis by A1,A4,A6,A8,A9;
  end;
end;
