
theorem MIN1:
  for a be non zero Nat,m,n be Nat holds
    min (a|^n,a|^m) = a|^(min (n,m))
  proof
    let a be non zero Nat,m,n be Nat;
      per cases;
      suppose
      A1: m >= n; then
      A2: min(m,n) = n by XXREAL_0:def 9;
      a|^n divides a|^m by A1,NEWTON:89; then
      a|^m >= a|^n by NAT_D:7;
      hence thesis by A2,XXREAL_0:def 9;
    end;
    suppose
      A1: n >= m; then
      A2: min(m,n) = m by XXREAL_0:def 9;
      a|^m divides a|^n by A1,NEWTON:89; then
      a|^n >= a|^m by NAT_D:7;
      hence thesis by A2,XXREAL_0:def 9;
    end;
  end;
