reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem Th46:
  a to_power 2 = a^2
proof
 now per cases;
    suppose
A1:   a>0;
      thus
      a to_power 2 = a to_power (1+1) .= a to_power 1 * a to_power 1 by A1,Th27
        .= a to_power 1 * a
        .= a^2;
    end;
    suppose
   a=0;
      hence thesis by Def2;
    end;
    suppose
A2:   a<0;
      reconsider l=1 as Integer;
      thus
      a to_power 2 = a #Z (l+l) by Def2
        .= a #Z l * a #Z l by A2,PREPOWER:44
        .= a * a #Z l by PREPOWER:35
        .= a^2 by PREPOWER:35;
    end;
  end;
  hence thesis;
end;
