reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th16: ::: EULER_1:14
  for a,b,c being Integer holds
  a <> 0 & c <> 0 & a,c are_coprime & b,c are_coprime implies a*b,c are_coprime
  proof
    let a,b,c be Integer;
    assume
A1: a <> 0 & c <> 0;
    assume a,c are_coprime;
    then
A2: |.a.|,|.c.| are_coprime by INT_2:34;
    assume b,c are_coprime;
    then |.b.|,|.c.| are_coprime by INT_2:34;
    then
A3: |.a.|*|.b.|,|.c.| are_coprime by A1,A2,EULER_1:14;
    |.a.|*|.b.| = |.a*b.| by COMPLEX1:65;
    hence thesis by A3,INT_2:34;
  end;
