reserve L for Abelian left_zeroed add-associative associative right_zeroed
              right_complementable distributive non empty doubleLoopStr;
reserve a,b,c for Element of L;
reserve R for non degenerated comRing;
reserve n,m,i,j,k for Nat;
 reserve D for Function of R, R;
 reserve x,y,z for Element of R;
reserve D for Derivation of R;
reserve s for FinSequence of the carrier of R;
reserve h for Function of R,R;
 reserve R for domRing;
 reserve f,g for Element of the carrier of Polynom-Ring R;
reserve a for Element of R;

theorem Th27:
     for i be Nat, p be Element of the carrier of Polynom-Ring R
     holds ((a|R)*'p).i = a*p.i
     proof
       let i be Nat, p be Element of the carrier of Polynom-Ring R;
       reconsider p1 = p as Polynomial of R;
       per cases;
         suppose
A1:        i = 0; then
           ((a|R)*'p1).i = (<%a%>*'p1).0 by Lm3
           .= (<%a,0.R%>*'p1).0 by POLYNOM5:43
           .= a*p1.i by A1,UPROOTS:37;
           hence thesis;
         end;
         suppose i <> 0; then
           consider j be Nat such that
A3:        i = j+ 1 by NAT_1:6;
           ((a|R)*'p1).i = (<%a%>*'p1).(j+1) by A3,Lm3
           .= (<%a,0.R%>*'p1).(j+1) by POLYNOM5:43
           .= a*p1.(j+1)+0.R*p1.j by UPROOTS:37
           .= a*p1.(j+1);
           hence thesis by A3;
          end;
       end;
