reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th42:
  n <> 0 implies Euler n <> 0
proof
  assume
A1: n <> 0;
  set X = {k where k is Element of NAT : n,k are_coprime & k >= 1 & k
  <= n};
A2: X c= finSeg n
  proof
    let x be object;
    assume x in X;
    then
    ex xx being Element of NAT st x = xx & n,xx are_coprime & xx >=
    1 & xx <= n;
    hence thesis by FINSEQ_1:1;
  end;
  assume Euler n = 0;
  then
A3: card X = 0 by EULER_1:def 1;
  reconsider X as finite set by A2;
  ex k st k in X
  proof
    take 1;
    n gcd 1 = 1 by NEWTON:51;
    then
A4: n,1 are_coprime by INT_2:def 3;
    1 <= n by A1,NAT_1:14;
    hence thesis by A4;
  end;
  hence contradiction by A3;
end;
