Uniqueness and origin of physics theorems

[simoncks]

After studying some of the proofs of equations, worries arises about their uniqueness.
1.Removal of integral
∫∇．Edτ = ∫(ρ/ε)dτ
→∇．E = Q/ε

2.Removal of (∇X)
∇XE = ∂B/∂t = ∂(∇XA)/∂t
→∇X(E+∂A/∂t)=0
→E+∂A/∂t=-∇V
→E=-∇V-∂A/∂t

Origin -
Certain physics equations/theorems have no (discrete physical/mathematical) proof, or my knowledge is not board enough.
1.F = dp/dt
2.E = F．s
3.Wave-particle duality
4.c (speed of light) is ALWAYS the same for ALL observers

Wouldn't the current theorems fail if the uniqueness fails/assumptions are not correct?

 

[andrien]

you can derive gauss law using maxwell fourth eqn and continuity eqn.
c2(∇×B)=j/ε0+∂E/∂t
taking the divergence of it and using ∇.j=-∂ρ/∂t one gets
∇.E=ρ/ε0