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

theorem Th15:
  for x, y being set st 0 < i & i < j holds {[i,x], [j,y]} is FinSubsequence
proof
  let x, y be set;
  assume that
A1: 0 < i and
A2: i < j;
  reconsider X = {[i,x],[j,y]} as Function by A2,GRFUNC_1:8;
A3: 0 + 1 <= i by A1,NAT_1:13;
  now
    let x be object;
    assume x in {i,j};
    then
A4: x = i or x = j by TARSKI:def 2;
    thus x in Seg j by A2,A3,A4,FINSEQ_1:3;
  end;
  then dom X = {i,j} & {i,j} c= Seg j by RELAT_1:10;
  hence thesis by FINSEQ_1:def 12;
end;
