reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;
reserve I,j for set;
reserve f,g for Function of I, NAT;
reserve J,K for finite Subset of I;

theorem Th29:
  sigma(k,1) = 1
proof
  set b = (EXP k)|NatDivisors 1;
  consider f be FinSequence of REAL such that
A1: Sum b = Sum f and
A2: f = b * (canFS (support b)) by UPROOTS:def 3;
  1 in NAT;
  then
A3: 1 in dom EXP k by FUNCT_2:def 1;
  1 in NatDivisors 1;
  then
A4: 1 in dom b by A3,RELAT_1:57;
  then
A5: b.1 = (EXP k).1 by FUNCT_1:47
    .= 1|^k by Def1
    .= 1;
  then for x being object holds x in support b iff x = 1
  by MOEBIUS1:41,TARSKI:def 1,PRE_POLY:def 7;
  then support b = {1} by TARSKI:def 1;
  then f = b * <*1*> by A2,FINSEQ_1:94
    .= <*b.1*> by A4,FINSEQ_2:34;
  then Sum b = 1 by A5,A1,RVSUM_1:73;
  hence sigma(k,1) = 1 by Def2;
end;
