
theorem N0141:
  for a,b,c,d be Integer holds a*b gcd c*d = 1 implies a gcd c = 1
  proof
    let a,b,c,d be Integer;
    |.a.| in NAT by INT_1:3; then
    reconsider k = |.a.| as Nat;
    |.b.| in NAT by INT_1:3; then
    reconsider m = |.b.| as Nat;
    |.c.| in NAT by INT_1:3; then
    reconsider l = |.c.| as Nat;
    |.d.| in NAT by INT_1:3; then
    reconsider n = |.d.| as Nat;
    assume a*b gcd c*d = 1; then
    1 = |.a*b.| gcd |.c*d.| by INT_2:34
    .= |.a.|*|.b.| gcd |.c*d.| by COMPLEX1:65
    .= |.a.|*|.b.| gcd |.c.|*|.d.| by COMPLEX1:65; then
    k*m, l*n are_coprime; then
    k,l are_coprime by NEWTON01:41;
    hence thesis by INT_2:34;
  end;
