Question: Trouble with nested loop and 2-level indices in dsolve

Below is a picture of my code and the errors that have arisen when I have tried to use a 2-level indice within a loop and dsolve. I will also paste in the code below.

restart;

with(PDEtools); umain := 1-exp(-y)+sum(A^i*u[i](y, z), i = 1 .. 4); vmain := -1+sum(A^i*v[i](y, z), i = 1 .. 4); wmain := sum(A^i*w[i](y, z), i = 1 .. 4); pde[main] := diff(vmain, y)+diff(wmain, z); pde[main[2]] := vmain*(diff(wmain, y))+wmain*(diff(wmain, z))-(diff(wmain, y$2));

for i to 1 do pde[i] := coeff(pde[main[2]], A^i) = 0; w[i](y, z) := sum(sol[i, k](y)*sin(k*z), k = 1 .. i); for j to i do sys__i__j := [-(diff(sol__i__j(y), y))-(diff(sol__i__j(y), y$2)) = coeff(pde[main[2]], sin(j*z)), sol__i__j(0) = 0, sol__i__j(infinity) = 0]*dsolve([-(diff(sol__i__j(y), y))-(diff(sol__i__j(y), y$2)) = coeff(pde[main[2]], sin(j*z)), sol__i__j(0) = 0, sol__i__j(infinity) = 0]); assign(op(0, sol__i__j*y)) = unapply(dsolve(sys__i__j), op(dsolve(sys__i__j))) end do end do