 reserve K,F,E for Field,
         R,S for Ring;

theorem Th7:
   S is RingExtension of R implies 0.(Polynom-Ring S) = 0.(Polynom-Ring R)
   proof
     assume S is R-extending Ring; then
A1:  R is Subring of S by Def1;
     thus
     0.(Polynom-Ring R) = 0_.(R) by POLYNOM3:def 10
     .= NAT --> 0.R by POLYNOM3:def 7
     .= NAT --> 0.S by A1,C0SP1:def 3
     .= 0_.(S) by POLYNOM3:def 7
     .= 0.(Polynom-Ring S) by POLYNOM3:def 10;
   end;
