reserve k for Nat;

theorem Th5:
  for i,n be Nat st i in Seg n holds i+1 in Seg (n+1)
proof
  let i,n be Nat;
  assume
A1: i in Seg n;
  then 1 <= i by FINSEQ_1:1;
  then 1+1 <= i+1 by XREAL_1:6;
  then
A2: 1 <= i+1 by XXREAL_0:2;
  i <= n by A1,FINSEQ_1:1;
  then i+1 <= n+1 by XREAL_1:6;
  hence thesis by A2,FINSEQ_1:1;
end;
