reserve a,b,c,k,l,m,n for Nat,
  i,j,x,y for Integer;

theorem Th14:
  a <> 0 & c <> 0 & a,c are_coprime & b,c are_coprime implies a*b,c are_coprime
proof
  assume that
A1: a <> 0 and
A2: c <> 0 and
A3: a,c are_coprime and
A4: b,c are_coprime;
A5: (a gcd c) = 1 by A3,INT_2:def 3;
A6: (a*b gcd c) divides c by NAT_D:def 5;
  ((a*b gcd c) gcd a) divides (a*b gcd c) by NAT_D:def 5;
  then ((a*b gcd c) gcd a) divides a & ((a*b gcd c) gcd a) divides c by A6,
NAT_D:4,def 5;
  then ((a*b gcd c) gcd a) divides 1 by A5,NEWTON:50;
  then
A7: ((a*b gcd c) gcd a) <= 0 + 1 by NAT_D:7;
  ((a*b gcd c) gcd a) <> 0 by A1,NEWTON:58;
  then ((a*b gcd c) gcd a) = 1 by A7,NAT_1:9;
  then (a*b gcd c) divides a*b & a,(a*b gcd c) are_coprime by
INT_2:def 3,NAT_D:def 5;
  then
A8: (a*b gcd c) divides b by Th13;
  (b gcd c) = 1 by A4,INT_2:def 3;
  then (a*b gcd c) divides 1 by A6,A8,NEWTON:50;
  then
A9: a*b gcd c <= 0 + 1 by NAT_D:7;
  (a*b) gcd c > 0 by A2,NEWTON:58;
  then a*b gcd c = 1 by A9,NAT_1:9;
  hence thesis by INT_2:def 3;
end;
