reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;

theorem Th21:
  n>0 & p is prime & p,q are_coprime implies not p divides (q mod p|^n)
proof
  assume that
A1: n>0 and
A2: p is prime and
A3: p,q are_coprime;
  n>=0+1 by A1,NAT_1:13;
  then p|^1 divides p|^n by NEWTON:89;
  then p divides p|^n;
  then
A4: p divides p|^n*(q div p|^n) by NAT_D:9;
  q=p|^n*(q div p|^n)+(q mod p|^n) by A2,NAT_D:2;
  hence thesis by A2,A3,A4,Lm6,NAT_D:8;
end;
