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

theorem Th37:
   for F being domRing, E being RingExtension of F
   for p being Polynomial of F for q being Polynomial of E
   for a being Element of F, b being Element of E
   st p = q & a = b holds a * p = b * q
   proof
     let F be domRing, E be RingExtension of F;
     let p be Polynomial of F; let q be Polynomial of E;
     let a be Element of F, b be Element of E;
     assume A1: p = q & a = b; then
A2:  (a|F) = (b|E) by FIELD_6:23;
     thus a * p = (a|F) *' p by FIELD_8:2
          .= (b|E) *' q by A2,A1,FIELD_4:17
          .= b * q by FIELD_8:2;
   end;
