reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th71:
  for f be real-valued FinSequence st i >1 & f.i >=0 holds
    (_sqrt f).i = sqrt (f.i)
proof
  let f be real-valued FinSequence such that
A1: i >1 & f.i >=0;
  set c = _sqrt f;
  len c = len f by Def11;
  then
A2: dom c = dom f by FINSEQ_3:29;
  per cases;
  suppose not i in dom f;
    then c.i = 0 & f.i = 0 by A2,FUNCT_1:def 2;
    hence thesis;
  end;
  suppose i in dom f;
    then
A3:(c.i)^2 = f.i & Re (c.i) >=0 & Im (c.i) >=0 by Def11,A1;
A4: sqrt (f.i)>=0 by A1,SQUARE_1:def 2;
A5: Re (sqrt (f.i)) = sqrt (f.i) & Im (sqrt (f.i)) = 0 by COMPLEX1:def 2;
    (sqrt (f.i))^2 = f.i by A1,SQUARE_1:def 2;
    hence thesis by A3,A5,Th2,A4;
  end;
end;
