reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th46:
  a>0 & n>=1 implies (n -Root a) #Z k = n -Root (a #Z k)
proof
  assume that
A1: a>0 and
A2: n>=1;
A3: n -Root a > 0 by A1,A2,Def2;
A4: m>=1 implies (n -Root a) |^ m = n -Root (a |^ m)
  proof
    assume that
A5: m>=1 and
A6: (n -Root a) |^ m <> n -Root (a |^ m);
A7: a |^ m > 0 by A1,Th6;
    then
A8: n -Root (a |^ m) >= 0 by A2,Def2;
A9: m -Root (n -Root (a |^ m)) = (n*m) -Root (a |^ m) by A2,A5,A7,Th25
      .= n -Root (m -Root (a |^ m)) by A2,A5,A7,Th25
      .= n -Root a by A1,A5,Lm2;
A10: m -Root ((n -Root a) |^ m) = n -Root a by A3,A5,Lm2;
A11: (n -Root a) |^ m >= 0 by A3,Th6;
    per cases by A6,XXREAL_0:1;
    suppose
      (n -Root a) |^ m < n -Root (a |^ m);
      hence contradiction by A5,A9,A11,Lm2;
    end;
    suppose
      (n -Root a) |^ m > n -Root (a |^ m);
      hence contradiction by A5,A9,A10,A8,Th28;
    end;
  end;
  per cases;
  suppose
A12: k>0;
    then |.k.|>0 by ABSVALUE:def 1;
    then
A13: |.k.|>=0+1 by NAT_1:13;
    thus (n -Root a) #Z k = (n -Root a) |^ |.k.| by A12,Def3
      .= n -Root (a |^ |.k.|) by A4,A13
      .= n -Root (a #Z k) by A12,Def3;
  end;
  suppose
A14: k<0;
    then |.k.|>0 by COMPLEX1:47;
    then
A15: |.k.|>=0+1 by NAT_1:13;
A16: a |^ |.k.| > 0 by A1,Th6;
    thus (n -Root a) #Z k = ((n -Root a) |^ |.k.|)" by A14,Def3
      .= (n -Root (a |^ |.k.|))" by A4,A15
      .= 1/n -Root (a |^ |.k.|)
      .= n -Root (1/a |^ |.k.|) by A2,A16,Th23
      .= n -Root ((a |^ |.k.|)")
      .= n -Root (a #Z k) by A14,Def3;
  end;
  suppose
A17: k=0;
    then n -Root (a #Z k) = n -Root 1 by Th34
      .= 1 by A2,Th20;
    hence thesis by A17,Th34;
  end;
end;
