Proof/

Lets define are mapping f:G to G/H+G/K by f(g)=(gH,GK). Need to show that this is a homomorphism. Obviously f is a well defined function. Operation Preserving- Let x,y be elements of G. Then f(xy)= (xyH,xyK)= (xHyH,xKyK)= (xH,XK),(yH,yK)=f(x)f(y) ( I need help justifying my step on this)

Thus f is a homomorphism. Since kerf ={g element of G| (gH,gK)=(H,K)}. thus f is injective (one to one).

So By 1st isomorphism theroem, G/Ker f is isomorphic to f(G). Since f(G) is a subgroup of G/H+G/K (since f is one to one), then G is isomorphic to a subgroup of G/H + G/K.

Does this seem right?