reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th44:
  h <> 0 implies rng (<*2,3*> ^ Sgm PrimeDivisors>3(h)) c= SetPrimes
  proof
    set X = PrimeDivisors>3(h);
    set F = Sgm X;
    set f = <*2,3*>;
    assume
A1: h <> 0;
A2: rng(f^F) = rng f \/ rng F by FINSEQ_1:31;
A3: rng f c= SetPrimes
    proof
      let x be object;
      assume x in rng f;
      then x in {2,3} by FINSEQ_2:127;
      then x = 2 or x = 3 by TARSKI:def 2;
      hence thesis by NEWTON:def 6,XPRIMES1:2,3;
    end;
    rng F c= SetPrimes
    proof
      let x be object;
      assume x in rng F;
      then x in X by A1,FINSEQ_1:def 14;
      then x is prime Nat by Th37;
      hence thesis by NEWTON:def 6;
    end;
    hence thesis by A2,A3,XBOOLE_1:8;
end;
