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
  for n being Nat holds
  n is odd iff ex k being Nat st n = 2*k+1
proof
  let n be Nat;
  hereby
    assume
A1: n is odd;
    then consider j being Integer such that
A2: n = 2*j+1 by Th1;
    now
      assume j < 0;
      then
A3:   2*j + 1 <= 2*0 by INT_1:7,XREAL_1:68;
      per cases by A3;
      suppose
        2*j+1 < 0;
        hence contradiction by A2;
      end;
      suppose
        2*j+1 = 0;
        then n = 2*0;
        hence contradiction by A1;
      end;
    end;
    then j in NAT by INT_1:3;
    then reconsider j as Nat;
    take j;
    thus n = 2*j+1 by A2;
  end;
  thus thesis;
end;
