reserve a,b,c for Integer;
reserve i,j,k,l for Nat;
reserve n for Nat;
reserve a,b,c,d,a1,b1,a2,b2,k,l for Integer;

theorem
  a,c are_coprime & b,c are_coprime implies a*b,c are_coprime
proof
  assume that
A1: a,c are_coprime and
A2: b,c are_coprime;
A4: ((a*b gcd c) gcd a) divides a by Def2;
A5: (a*b gcd c) divides c by Def2;
  ((a*b gcd c) gcd a) divides (a*b gcd c) by Def2;
  then ((a*b gcd c) gcd a) divides c by A5,Lm2;
  then ((a*b gcd c) gcd a) divides 1 by A1,A4,Def2;
  then a,(a*b gcd c) are_coprime by INT_1:9;
  then (a*b gcd c) divides b by Th21,Th25;
  then (a*b gcd c) divides 1 by A2,A5,Def2;
  hence thesis by INT_1:9;
end;
