reserve i,j,k,n,m for Nat,
        b,b1,b2 for bag of n;

theorem Th13:
  for L being well-unital non trivial doubleLoopStr
    for x being Function of n, L, y be Function of n+1,L st y|n = x
      holds
  eval(b,x) = eval(b bag_extend 0,y)
proof
  let L be well-unital non trivial doubleLoopStr;
  let x be Function of n, L, y be Function of n+1,L such that
A1: y|n = x;
  set S=SgmX(RelIncl n, support b);
  set B=b bag_extend 0;
  set S1=SgmX(RelIncl (n+1), support B);
  consider P be FinSequence of L such that
A2: len P = len S & eval(b,x) = Product P and
A3: for i be Element of NAT st
  1 <= i & i <= len P holds
  P/.i = power(L).((x * S)/.i,(b * S)/.i) by POLYNOM2:def 2;
  consider P1 be FinSequence of L such that
A4: len P1 = len S1 & eval(B,y) = Product P1 and
A5: for i be Element of NAT st
  1 <= i & i <= len P1 holds
  P1/.i = power(L).((y * S1)/.i,(B * S1)/.i) by POLYNOM2:def 2;
A6:S=S1 by Th12;
A7: rng S c= n;
A8:y*S = y*((id n)*S) by A7,RELAT_1:53
  .= y*(id n)*S by RELAT_1:36
  .= x*S by RELAT_1:65,A1;
A9:b=(0,n)-cut B by Th5
  .= B|n by NAT_1:11,Th3;
A10:B*S = B*((id n)*S) by A7,RELAT_1:53
  .= B*(id n)*S by RELAT_1:36
  .=b*S by A9,RELAT_1:65;
  for i be Nat st 1<= i <= len P holds P.i=P1.i
  proof
    let i be Nat such that
A11: 1<=i <= len P;
    reconsider I=i as Element of NAT by ORDINAL1:def 12;
A12:i in dom P & i in dom P1 by A2,A4,A6,A11,FINSEQ_3:25;
    then P/.I = P.i by PARTFUN1:def 6;
    hence P.i = power(L).((y * S1)/.i,(B * S1)/.i) by A3,A11,A6,A8,A10
    .= P1/.I by A5,A11,A2,A4,A6
    .= P1.i by A12,PARTFUN1:def 6;
  end;
  hence thesis by A2,A4,A6,FINSEQ_1:14;
end;
