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

theorem Th7:
  for m,n being Nat holds
  m divides n*p implies m divides n or ex z being Nat st m = z*p & z divides n
  proof
    let m,n be Nat;
    assume m divides n*p;
    then consider a,b being Nat such that
A1: a divides n and
A2: b divides p and
A3: m = a * b by NUMBER12:14;
    assume
A4: not m divides n;
    per cases by A2,INT_2:def 4;
    suppose b = 1;
      hence thesis by A1,A3,A4;
    end;
    suppose
A5:   b = p;
      take a;
      thus m = a*p by A3,A5;
      thus a divides n by A1;
    end;
  end;
