reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th7:
  for m,n being Nat holds m <> 0 & m divides n mod m implies m divides n
proof
  let m,n be Nat;
  assume that
A1: m <> 0 and
A2: m divides n mod m;
  consider x be Nat such that
A3: n mod m = m * x by A2,NAT_D:def 3;
  n mod m + m * (n div m) = m * (x + (n div m)) by A3;
  then n = m * (x + (n div m)) by A1,NAT_D:2;
  hence thesis;
end;
