reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;
reserve n, m for non zero Nat;

theorem Th52:
  support pfexp n = {} implies n = 1
proof
  assume that
A1: support pfexp n = {} and
A2: n <> 1;
  n >= 0+1 by NAT_1:13;
  then n > 1 by A2,XXREAL_0:1;
  then n >= 1+1 by NAT_1:13;
  then consider p being Element of NAT such that
A3: p is prime and
A4: p divides n by INT_2:31;
  p > 1 by A3;
  then p |-count n <> 0 by A4,Th27;
  then (pfexp n).p <> 0 by A3,Def8;
  hence contradiction by A1,PRE_POLY:def 7;
end;
