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

theorem Th45:
  for a,b being non zero Nat holds support pfexp a c= support pfexp (a*b)
proof
  let a, b be non zero Nat;
  set f = pfexp a;
  set h = pfexp (a*b);
  let x be object;
  assume
A1: x in support f;
  then reconsider x as Prime by Th34;
A2: f.x = x |-count a by Def8;
  f.x <> 0 by A1,PRE_POLY:def 7;
  then
A3: x divides x |^ (x |-count a) by A2,Th3;
A4: x <> 1 by INT_2:def 4;
  then x |^ (x |-count a) divides a by Def7;
  then x divides a by A3,NAT_D:4;
  then x |^ 1 divides a;
  then
A5: x |^ (0+1) divides a*b by NAT_D:9;
  h.x = x |-count (a*b) by Def8;
  then h.x <> 0 by A4,A5,Def7;
  hence thesis by PRE_POLY:def 7;
end;
