reserve x,y,a,b,c,p,q for Real;
reserve m,n for Element of NAT;

theorem Th4:
  a>0 & n is even & n >= 1 & x |^ n = a implies x = n-root a or x = -n-root a
proof
  assume that
A1: a>0 and
A2: n is even and
A3: n >= 1;
  assume
A4: x |^ n=a;
then A5: x<>0 by A1,A3,NEWTON:11;
  now
    per cases by A5;
    case x>0;
      hence thesis by A4,A3,POWER:4;
    end;
    case x<0;
      then
A6:   -x>0 by XREAL_1:58;
      n-root a=n-root((-x)|^ n) by A2,A4,POWER:1;
      then (-1)*n-root a=(-1)*(-x) by A3,A6,POWER:4;
      hence thesis;
    end;
  end;
  hence thesis;
end;
