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

theorem Th9:
   S is RingExtension of R implies 1.(Polynom-Ring S) = 1.(Polynom-Ring R)
   proof
     assume
A1:  S is R-extending Ring; then
A2:  R is Subring of S by Def1;
     thus 1.(Polynom-Ring R) = 1_.(R) by POLYNOM3:def 10
     .= 0_.(R)+*(0,1.R) by POLYNOM3:def 8
     .= 0_.(R)+*(0,1.S) by A2,C0SP1:def 3
     .= 0_.(S)+*(0,1.S) by A1,Th8
     .= 1_.(S) by POLYNOM3:def 8
     .= 1.(Polynom-Ring S) by POLYNOM3:def 10;
   end;
