reserve a,b,c,k,l,m,n for Nat,
  i,j,x,y for Integer;

theorem Th19:
  n > 1 implies Euler n <= n - 1
proof
  set X={kk where kk is Element of NAT : n,kk are_coprime & kk >= 1 &
  kk <= n};
  X c= Seg n
  proof
    let x be object;
    assume x in X;
    then ex k be Element of NAT st k = x & n,k are_coprime & k >= 1 & k
    <= n;
    hence thesis by FINSEQ_1:1;
  end;
  then reconsider X as finite set;
  assume
A1: n > 1;
  then 0 in {l where l is Nat: l < n};
  then 0 in n by AXIOMS:4;
  then
A2: card(n \ {0}) = card n - card {0} by Th4;
A3: X c= n \ {0}
  proof
    let x be object;
    assume x in X;
    then consider k be Element of NAT such that
A4: k = x and
A5: n,k are_coprime and
A6: k >= 1 and
A7: k <= n;
    not n,n are_coprime by A1,Th1;
    then k < n by A5,A7,XXREAL_0:1;
    then k in {l where l is Nat: l < n};
    then
A8: k in n by AXIOMS:4;
    not k in {0} by A6,TARSKI:def 1;
    hence thesis by A4,A8,XBOOLE_0:def 5;
  end;
  card n = n & card {0} = 1 by CARD_1:30;
  hence thesis by A3,A2,NAT_1:43;
end;
