reserve a,b for Complex;
reserve z for Complex;
reserve n0 for non zero Nat;

theorem Th10:
  z|^2 = a iff z = 2-root a or z = -2-root a
proof
A1: a = (2-root a)|^2 by Th7
    .= (2-root a)*(2-root a) by Th1;
  hereby
    assume z|^2 = a;
    then
A2: z*z = a by Th1;
    assume not z = 2-root a;
    then
A3: (z - 2-root a) <> 0;
    assume not z = -2-root a;
    then (z + 2-root a) <> 0;
    then (z - 2-root a)*(z + 2-root a) <> 0 by A3;
    hence contradiction by A1,A2;
  end;
  assume z = 2-root a or z = -2-root a;
  then z*z - (2-root a)*(2-root a) = 0;
  hence z|^2 = a by A1,Th1;
end;
