reserve n, k, r, m, i, j for Nat;

theorem Th18:
  for i being Element of NAT, y being set, f being FinSubsequence
  st f = { [1,y] } holds Shift(f,i) = { [1+i,y] }
proof
  let i be Element of NAT, y be set, f be FinSubsequence;
  set g = Shift(f,i);
  assume
A1: f = { [1,y] };
  then card f = 1 by CARD_2:42;
  then card (Shift(f,i)) = 1 by VALUED_1:42;
  then
A2: ex x being object st Shift(f,i) = {x} by CARD_2:42;
A3: dom f = {1} by A1,RELAT_1:9;
  dom g = {1+i}
  proof
    hereby
      let x be object;
      assume x in dom g;
      then x in {o+i where o is Nat : o in dom f } by VALUED_1:def 12;
      then consider w being Nat such that
A4:   w+i = x and
A5:   w in dom f;
      w = 1 by A3,A5,TARSKI:def 1;
      hence x in {1+i} by A4,TARSKI:def 1;
    end;
    let x be object;
    assume x in {1+i};
    then
A6: x = 1 + i by TARSKI:def 1;
    1 in dom f by A3,TARSKI:def 1;
    then x in {o+i where o is Nat : o in dom f } by A6;
    hence thesis by VALUED_1:def 12;
  end;
  then
A7: 1 + i in dom g by TARSKI:def 1;
  1 in dom f by A3,TARSKI:def 1;
  then g.(1+i) = f.1 by VALUED_1:def 12
    .= y by A1,GRFUNC_1:6;
  then [1+i,y] in g by A7,FUNCT_1:def 2;
  hence thesis by A2,TARSKI:def 1;
end;
