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

theorem Th46:
  for a, b being non zero Nat holds
  support pfexp (a*b) = support pfexp a \/ support pfexp b
proof
  let a, b be non zero Nat;
  set f = pfexp a;
  set g = pfexp b;
  set h = pfexp (a*b);
  thus support h c= support f \/ support g
  proof
    let x be object;
    assume
A1: x in support h;
    then reconsider x as Prime by Th34;
A2: h.x <> 0 by A1,PRE_POLY:def 7;
A3: x <> 1 by INT_2:def 4;
A4: h.x = x |-count (a*b) & x |^ (x |-count(a*b)) = (x |^ (x |-count a)) *
    (x |^ (x |-count b)) by Def8,Th29;
    per cases by A2,A4,Th3,NEWTON:80;
    suppose
      x divides x |^ (x |-count a);
      then x divides a by A3,Th26;
      then f.x <> 0 by Th38;
      then x in support f by PRE_POLY:def 7;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      x divides x |^ (x |-count b);
      then x divides b by A3,Th26;
      then g.x <> 0 by Th38;
      then x in support g by PRE_POLY:def 7;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  support f c= support h & support g c= support h by Th45;
  hence thesis by XBOOLE_1:8;
end;
