
theorem extsubst:
for F being Field
for E being FieldExtension of F
for p,q being Polynomial of F
for p1,q1 being Polynomial of E
st p1 = p & q1 = q holds Subst(p1,q1) = Subst(p,q)
proof
let F be Field, E be FieldExtension of F;
let p,q be Polynomial of F; let p1,q1 be Polynomial of E;
assume AS: p1 = p & q1 = q;
consider f being FinSequence of the carrier of Polynom-Ring F such that
A1: Subst(p,q) = Sum f & len f = len p &
    for n being Element of NAT st n in dom f holds f.n = p.(n-'1)*(q`^(n-'1))
    by POLYNOM5:def 6;
consider g being FinSequence of the carrier of Polynom-Ring E such that
A2: Subst(p1,q1) = Sum g & len g = len p1 &
    for n being Element of NAT st n in dom g holds g.n = p1.(n-'1)*(q1`^(n-'1))
    by POLYNOM5:def 6;
    p is Element of the carrier of Polynom-Ring F &
    p1 is Element of the carrier of Polynom-Ring E by POLYNOM3:def 10; then
    deg p1 = deg p by AS,FIELD_4:20; then
    deg p = len p1 - 1 by HURWITZ:def 2; then
H1: len p - 1 = len p1 - 1 by HURWITZ:def 2;
A3: f = g
    proof
    B1: dom f = Seg(len p1) by H1,A1,FINSEQ_1:def 3
             .= dom g by A2,FINSEQ_1:def 3;
    now let n be Element of NAT;
      assume B2: n in dom f;
      n -' 1 is Element of NAT by ORDINAL1:def 12; then
      B3: q`^(n-'1) = q1`^(n-'1) & p.(n-'1) = p1.(n-'1) by AS,multi00;
      B4: ((p.(n-'1))|F) = ((p1.(n-'1))|E) by AS,FIELD_6:23;
      thus f.n = p.(n-'1) * (q`^(n-'1)) by B2,A1
              .= ((p.(n-'1))|F) *' (q`^(n-'1)) by FIELD_8:2
              .= ((p1.(n-'1))|E) *' (q1`^(n-'1)) by B3,B4,FIELD_4:17
              .= p1.(n-'1) * (q1`^(n-'1)) by FIELD_8:2
              .= g.n by A2,B1,B2;
      end;
    hence thesis by B1;
    end;
    Polynom-Ring F is Subring of Polynom-Ring E by FIELD_4:def 1;
hence thesis by A1,A2,A3,FIELD_4:2;
end;
