theorem Th3:
  x>=0 & y>=0 & d>0 & x|^d=y|^d implies x=y
proof
  assume that
A1: x>=0 and
A2: y>=0 and
A3: d>0 and
A4: x|^d=y|^d;
A5: d>=1 by A3,NAT_1:14;
  then x = d -Root (x |^ d) by A1,PREPOWER:19
    .= y by A2,A4,A5,PREPOWER:19;
  hence thesis;
end;
