theorem Th15:
  a,m are_coprime implies ex x being Integer st (a*x-b) mod m = 0
proof
  assume a,m are_coprime;
  then a gcd m = 1 by INT_2:def 3;
  then consider s,t being Integer such that
A1: 1=s*a+t*m by NAT_D:68;
  take b*s;
  (a*b*s-b) mod m = ((a*s-1)*b) mod m .= ((-(m*t))*b) mod m by A1
    .= (0+m*((-t)*b)) mod m
    .= 0 mod m by NAT_D:61
    .= 0 by Th12;
  hence thesis;
end;
