
theorem
  for k, n, i being Nat st 1 <= k holds i in Seg n iff k * i in Seg (k * n)
proof
  let k, n, i be Nat;
  assume
A1: 1 <= k;
  hereby
    assume
A2: i in Seg n;
    then i <= n by FINSEQ_1:1;
    then
A3: k * i <= k * n by NAT_1:4;
    1 <= i by A2,FINSEQ_1:1;
    then 1 * 1 <= k * i by A1,XREAL_1:66;
    hence k * i in Seg (k * n) by A3,FINSEQ_1:1;
  end;
  assume
A4: k * i in Seg (k * n);
  then 0 < i by FINSEQ_1:1;
  then
A5: 0 + 1 <= i by NAT_1:13;
  k * i <= k * n by A4,FINSEQ_1:1;
  then i <= n by A1,XREAL_1:68;
  hence thesis by A5,FINSEQ_1:1;
end;
