
theorem
for p being Prime
for n,a,b being Nat
st p divides a & not p divides b & n = a/b holds p divides n
proof
let p be Prime, n,a,b be Nat;
assume that A0: p divides a & not p divides b and
B2: n = a/b;
per cases;
suppose B1: b is non zero;
consider x being Nat such that A1: p * x = a by A0,NAT_D:def 3;
A2: p * (x/b) = (p * x) / b;
consider u being Nat such that A3: b * u = p * x
by B2,A1,Lm3b,B1,NAT_D:def 3;
p divides u by A0,A3,NAT_D:def 3,NEWTON:80; then
consider v being Nat such that A4: p * v = u by NAT_D:def 3;
(b * v) * p = x * p & p <> 0 by A3,A4;
then b divides x by XCMPLX_1:5,NAT_D:def 3;
then x/b is Nat by B1,Lm3b;
hence thesis by B2,A1,A2,NAT_D:def 3;
end;
suppose b is zero;
  then n = 0 by B2;
  hence thesis by INT_2:12;
end;
end;
