reserve a,b,c,d,x,j,k,l,m,n,o,xi,xj for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem MM1:
  for a,b be non negative Real holds
  max(a|^n,b|^n) = (max(a,b))|^n & min (a|^n,b|^n) = (min(a,b))|^n
  proof
    let a,b be non negative Real;
    per cases;
    suppose
      a >= b; then
      a|^n = max(a|^n,b|^n) & b|^n = min(a|^n,b|^n) &
      a = max(a,b) & b = min(a,b) by NEWTON02:41,XXREAL_0:def 9,def 10;
      hence thesis;
    end;
    suppose
      a < b; then
      b|^n = max(a|^n,b|^n) & a|^n = min(a|^n,b|^n) &
      b = max(a,b) & a = min(a,b) by NEWTON02:41,XXREAL_0:def 9,def 10;
      hence thesis;
    end;
  end;
