
theorem
  for a,b be non zero Integer holds
  Parity (a+b) = (min (Parity a,Parity b))*
    Parity ((a+b)/min(Parity a,Parity b))
  proof
    let a,b be non zero Integer;
    A1: min (Parity a,Parity b) = Parity a or
      min (Parity a,Parity b) = Parity b by XXREAL_0:def 9;
    min (Parity a,Parity b) divides (Parity a)
      by A1,XXREAL_0:def 9,PEPIN31; then
    consider c be Nat such that
    A2: Parity a = min (Parity a,Parity b)*c by NAT_D:def 3;
    min (Parity a,Parity b) divides (Parity b)
      by A1,XXREAL_0:def 9,PEPIN31; then
    consider d be Nat such that
    A3: Parity b = min (Parity a,Parity b)*d by NAT_D:def 3;
    (Parity a)/min (Parity a,Parity b) = c &
      (Parity b)/min (Parity a,Parity b) = d by A2,A3,XCMPLX_1:89; then
    A4: (Oddity a)*c = a/min (Parity a,Parity b) &
      (Oddity b)*d = b/min (Parity a,Parity b) by XCMPLX_1:98;
    a + b = (Oddity a)*(min(Parity a,Parity b)*c) +
      (Oddity b)*(min(Parity a,Parity b)*d) by A2,A3
    .= min(Parity a,Parity b)*((Oddity a)*c + (Oddity b)*d); then
    Parity (a+b)
    = Parity (min(Parity a,Parity b))*Parity((Oddity a)*c + (Oddity b)*d)
      by ILP
    .= min (Parity a, Parity b) * Parity((Oddity a)*c + (Oddity b)*d) by A1;
    hence thesis by A4,XCMPLX_1:62;
  end;
