reserve a,b,m,x,n,l,xi,xj for Nat,
  t,z for Integer;

theorem Th3:
  a <> 0 & b <> 0 & m <> 0 & a,m are_coprime & b,m are_coprime implies
  m,a*b mod m are_coprime
proof
  assume that
A1: a <> 0 and
A2: b <> 0 and
A3: m <> 0 and
A4: a,m are_coprime & b,m are_coprime;
  a*b,m are_coprime by A1,A3,A4,EULER_1:14;
  then
A5: a*b gcd m = 1;
  consider t being Nat such that
A6: a*b = m*t+(a*b mod m) and
  (a*b mod m) < m by A3,NAT_D:def 2;
  a*b <> a*0 by A1,A2;
  then a*b+(-t)*m gcd m = a*b gcd m by EULER_1:16;
  hence thesis by A6,A5;
end;
