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

theorem Th44:
  for a, b being non zero Nat st a,b are_coprime holds
  support pfexp a misses support pfexp b
proof
  let a, b be non zero Nat;
  set f = pfexp a;
  set g = pfexp b;
  assume a,b are_coprime;
  then
A1: a gcd b = 1;
  assume support f meets support g;
  then consider x being object such that
A2: x in support f and
A3: x in support g by XBOOLE_0:3;
  reconsider x as Prime by A2,Th34;
A4: f.x = x |-count a by Def8;
A5: g.x = x |-count b by Def8;
  g.x <> 0 by A3,PRE_POLY:def 7;
  then
A6: x divides x |^ (x |-count b) by A5,Th3;
A7: x <> 1 by INT_2:def 4;
  then x |^ (x |-count b) divides b by Def7;
  then
A8: x divides b by A6,NAT_D:4;
  f.x <> 0 by A2,PRE_POLY:def 7;
  then
A9: x divides x |^ (x |-count a) by A4,Th3;
  x |^ (x |-count a) divides a by A7,Def7;
  then x divides a by A9,NAT_D:4;
  then x divides 1 by A1,A8,NAT_D:def 5;
  hence contradiction by A7,WSIERP_1:15;
end;
