 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem
   for F being domRing, E being domRingExtension of F
   for p being Polynomial of F
   for a being Element of F, x,b being Element of E st b = a
   holds Ext_eval(a*p,x) = b * Ext_eval(p,x)
   proof
     let F be domRing, E be domRingExtension of F, p be Polynomial of F;
     let a be Element of F, x,b being Element of E;
     assume A1: b = a;
     reconsider p1 = p as Element of the carrier of Polynom-Ring F
     by POLYNOM3:def 10;
     the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
     by FIELD_4:10; then
     reconsider q1 = p1 as Element of the carrier of Polynom-Ring E;
     reconsider q0 = q1 as Polynomial of E;
A2:  a * p1 is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
     b * q1 is Element of the carrier of Polynom-Ring E by POLYNOM3:def 10;
       then
     Ext_eval(a*p1,x) = eval(b*q1,x) by A2,FIELD_4:26,A1,Th37
     .= b * eval(q0,x) by RING_5:7
     .= b * Ext_eval(p1,x) by FIELD_4:26;
     hence thesis;
   end;
