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
  m<>0 & not (a gcd m) divides b implies not ex x being Integer st (a*x-
  b) mod m = 0
proof
  assume that
A1: m<>0 and
A2: not (a gcd m) divides b;
  given y such that
A3: (a*y-b) mod m = 0;
  (a gcd m) divides m by INT_2:21;
  then
A4: ex i being Integer st m=(a gcd m)*i;
  (a*y-b) mod m = 0 mod m by A3,Th12;
  then (a*y-b),0 are_congruent_mod m by A1,NAT_D:64;
  then (a*y-b),0 are_congruent_mod (a gcd m) by A4,INT_1:20;
  then
A5: (a gcd m) divides (a*y-b-0);
  (a gcd m) divides a*y by INT_2:2,21;
  hence contradiction by A2,A5,Lm5;
end;
