reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th38:
  Euler 0 = 0
  proof
    set A = {k where k is Element of NAT: 0,k are_coprime & k >= 1 & k <= 0};
    assume Euler 0 <> 0;
    then card A <> 0 by EULER_1:def 1;
    then consider a being object such that
A1: a in A by XBOOLE_0:7,CARD_1:27;
    ex k being Element of NAT st k = a & 0,k are_coprime & k >= 1 & k <= 0
    by A1;
    hence contradiction;
  end;
