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 POW2:
  for a,b be Real, n be odd Nat holds a|^n = b|^n iff a = b
  proof
    let a,b be Real, n be odd Nat;
    per cases;
    suppose
  A1: a >= 0;
      per cases;
      suppose b >= 0;
        hence thesis by A1,POW1;
      end;
      suppose b < 0;
        hence thesis by A1;
      end;
    end;
    suppose
  A2: a < 0;
      per cases;
      suppose b < 0; then
        reconsider k = -b as positive Real;
        reconsider l = -a as positive Real by A2;
    B1: (-a)|^n = -(a|^n) & (-b)|^n = -(b|^n) by POWER:2;
        k|^n = l|^n iff k = l by POW1;
        hence thesis by B1;
      end;
      suppose b >= 0;
        hence thesis by A2;
      end;
    end;
  end;
