The force of friction at any point is proportional to the local pressure at that point*, i.e, doubling the pressure there doubles the friction.
The net "pressure force" F (or applied normal force) is gained by summing up the pressures at all points:
F=\int_{S}pdA
where S is the surface we sum over, p is the local pressure, and dA the differential area element.
Now, the proportionality factor between the force of friction and the local pressure will most likely only vary significantly if the contact surface properties themselves vary significantly**.
So, if the contact surface can be regarded as relatively homogenous in material properties, the local force of friction can be written as:
dF_{fric}=\mu{p}dA
where dF_{fric} is the local friction force
and \mu the constant proportionality factor.
Thus, the net friction force is given by:
F_{fric}=\int_{S}dF_{fric}=\int_{S}\mu{p}dA=\mu\int_{S}pdA=\mu{F}
Thus, the force of friction is strictly proportional to the applied normal force, which again equals the normal force N from the surface, since the objects don't slide into each other.
Thus, the friction force is proportional to N.
In particular, in so far as the applied force remains the same, whereas the contact surface is changed, no change will be seen in the net friction force.
It is LOCALLY stronger, just as the local pressure is stronger, but the total surface is less, so the whole balances neatly.
*This is our starting HYPOTHESIS, that has ample empirical verification.
** The friction coefficient happens to be EXTREMELY sensitive to a lot of factors: temperature, presence of material impurities and so on.
This by no means reduce the validity of the argument below, but does, indeed, limit the usefulness of the model.
