Provided we have a branch ab and Vd>Vc plus Va=Vc,Vb=Vd ,Is it possible to have certain node (denoted by "m") in the branch ab whose voltage Vm is bigger than Vd??

originally ,I wanna prove (TT:current can only flow from c to d)by proving its wrong when its direction is converse(d to c) ,then I label corresponding current direction as shown above. Afterwards,I realize where it goes wrong: (A:current doesn not flow out of d ) .Next, I attempted to prove that A is necessarily wrong through proving that (opposite A) is wrong. Providing (opposite A) is true,we can draw a conclusion that certain node (denoted by "m") in the branch AB whose voltage Vm is bigger than Vd ， Whether this conclusion is right or wrong bothers me? Plus,whats the equivalant circuit for between node A and node B and how to draw it ?

 

There so much wrong just in the first clause in the first sentence.

Provided we have a branch ab

Okay, so far so god.

and Vd>Vc

On what basis is this assertion based, given the location of the only power source, the 12 V battery, and its orientation?

plus Va=Vc,Vb=Vd

And these claims are based on what?

They required that there be no current in either the 4 Ω or the 2 Ω resistors, and hence no current in the battery. Does this make sense?

If true, it also requires that Vb > Va, but the battery imposes the constraint that Va = Vb + 12 V. Does this make sense?

,Is it possible to have certain node (denoted by "m") in the branch ab whose voltage Vm is bigger than Vd??

Vm, Vn, and Vd are the SAME node. What do you know about the voltage at all locations on the same node in a circuit?

(A:current doesn not flow out of d )

What does this mean? KCL dictates that any current that flows into a node must flow out of it (and vice versa). So the only way this can be true is if no current is flowing through that node at all.

.Next, I attempted to prove that A is necessarily wrong through proving that (opposite A) is wrong. Providing (opposite A) is true,we can draw a conclusion that certain node (denoted by "m") in the branch AB whose voltage Vm is bigger than Vd ，

What conclusion?

Whether this conclusion is right or wrong bothers me? Plus,whats the equivalant circuit for between node A and node B and how to draw it ?

Well, draw the equivalent circuit. Draw a horizontal line and call that Node A. Draw another horizontal line a ways below that and call that Node B. Now look at each of those three 9 Ω resistors and if they are connected directly to Node A and Node B, draw them connected between your two horizontal lines in your diagram.

It might help to color each node using a separate color.