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

theorem Th42:
  a <> 0 implies support pfexp (p|^a) = {p}
proof
  set f = pfexp (p|^a);
  assume a <> 0;
  then p divides p|^a by Th3;
  then
A1: f.p <> 0 by Th38;
  thus support f c= {p}
  proof
    let x be object;
    assume
A2: x in support f;
    then reconsider x as Prime by Th34;
A3: x <> 1 & f.x = x |-count (p|^a) by Def8,INT_2:def 4;
    f.x <> 0 by A2,PRE_POLY:def 7;
    then x divides p|^a by A3,Th27;
    then x = p by Th6;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {p};
  then x = p by TARSKI:def 1;
  hence thesis by A1,PRE_POLY:def 7;
end;
