
theorem Lm9:
for F being Field,
    E being FieldExtension of F,
    K being F-extending FieldExtension of E
for h being F-fixing Monomorphism of E,K
for T being non empty finite Subset of E
for b being bag of (card T)
for x being T-evaluating Function of (card T),E
holds h.eval(b,x) in the carrier of RAdj(F,h.:T)
proof
let F be Field, E be FieldExtension of F,
    K be F-extending FieldExtension of E;
let h be F-fixing Monomorphism of E,K, T be non empty finite Subset of E;
let b be bag of (card T), x be T-evaluating Function of (card T),E;
defpred P[Nat] means
   for b being bag of (card T) st card(support b) = $1
   for x being T-evaluating Function of (card T),E
   holds h.eval(b,x) in the carrier of RAdj(F,h.:T);
set n = card T, R = RAdj(F,h.:T);
H0: 0.K = 0.R & 1.K = 1.R by FIELD_6:def 4;
A: P[0]
   proof
   now let b be bag of n;
     assume A0: card(support b) = 0;
     let x be T-evaluating Function of n,E;
     now let o be object;
       assume o in n;
       not o in support b by A0;
       hence b.o = {} by PRE_POLY:def 7 .= (EmptyBag n).o by PBOOLE:5;
       end; then
     b = EmptyBag n by PBOOLE:def 10; then
     h.eval(b,x) = h.(1_E) by POLYNOM2:14 .= 1_K by GROUP_1:def 13;
     hence h.eval(b,x) in the carrier of RAdj(F,h.:T) by H0;
     end;
   hence thesis;
   end;
B: now let k be Nat;
   assume IV: P[k];
   now let b be bag of n;
     assume B0: card(support b) = k+1;
     let x be T-evaluating Function of n,E;
     set a = the Element of support b;
     B3: support b <> {} by B0; then
     B1: a in support b & support b c= dom b & dom b c= n by PRE_POLY:37; then
     reconsider a as Element of n;
     set b1 = b \ a; set b2 = ({a},b.a)-bag;
     B2: support b1 = support b \ {a} by RING_5:29;
     {a} c= support b by B1,TARSKI:def 1; then
     card(support b1)
        = card(support b) - card {a} by B2,CARD_2:44
       .= (k + 1) - 1 by B0,CARD_2:42; then
     reconsider u = h.eval(b1,x) as Element of RAdj(F,h.:T) by IV;
     b.a <> 0 by B3,PRE_POLY:def 7; then
     support b2 = {a} by UPROOTS:8; then
     reconsider v = h.eval(b2,x) as Element of RAdj(F,h.:T) by Lm10;
     h.eval(b,x) = h.eval(b1 + b2,x) by RING_5:30
                .= h.(eval(b1,x) * eval(b2,x)) by POLYNOM2:16
                .= h.eval(b1,x) * h.eval(b2,x) by GROUP_6:def 6
                .= u * v by FIELD_6:16;
     hence h.eval(b,x) in the carrier of RAdj(F,h.:T);
     end;
   hence P[k+1];
   end;
C: for k being Nat holds P[k] from NAT_1:sch 2(A,B);
consider n being Nat such that D: card(support b) = n;
thus thesis by C,D;
end;
