reserve x, y, z, E, E1, E2, E3 for set,
  sE for Subset-Family of E,
  f for Function of E, E,
  k, l, m, n for Nat;

theorem Th1:
  for i being Integer holds i is odd iff ex j being Integer st i = 2*j+1
proof
  let i be Integer;
  hereby
    consider k such that
A1: i = k or i = -k by INT_1:2;
    consider m being Element of NAT such that
A2: k = 2*m or k = 2*m+1 by SCHEME1:1;
    assume
A3: i is odd;
    assume
A4: for j being Integer holds i <> 2*j+1;
    per cases by A1,A2;
    suppose
      i = k & k = 2*m;
      hence contradiction by A3,Lm1;
    end;
    suppose
      i = -k & k = 2*m;
      then i = 2*(-m);
      hence contradiction by A3,Lm1;
    end;
    suppose
      i = k & k = 2*m+1;
      hence contradiction by A4;
    end;
    suppose
      i = -k & k = 2*m+1;
      then i = 2*-(m+1)+1;
      hence contradiction by A4;
    end;
  end;
  given j being Integer such that
A5: i = 2*j+1;
  given k being Integer such that
A6: i = 2*k;
  1 = 2*(k - j) by A5,A6;
  hence contradiction by INT_1:9;
end;
