reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th68:
  for m be Ordinal for b be bag of m
    for xc be Function of m,F_Complex,xr be Function of m,F_Real st
      xc = xr
  holds eval(b,xc) = eval(b,xr)
proof
  let m be Ordinal;
  let b be bag of m;
  let xc be Function of m,F_Complex, xr be Function of m,F_Real such that
A1:xc = xr;
  reconsider FC = F_Complex,FR = F_Real as Field;
  reconsider xc as Function of m,FC;
  reconsider xr as Function of m,FR;
  set S=SgmX(RelIncl m, support b);
  consider yC being FinSequence of FC such that
A2: len yC = len S & eval(b,xc) = Product yC and
A3: for i being Element of NAT st 1 <= i & i <= len yC holds
  yC/.i = power(FC).((xc * S)/.i,(b * S)/.i) by POLYNOM2:def 2;
  consider yR being FinSequence of FR such that
A4: len yR = len S & eval(b,xr) = Product yR and
A5: for i being Element of NAT st 1 <= i & i <= len yR holds
  yR/.i = power(FR).((xr * S)/.i,(b * S)/.i) by POLYNOM2:def 2;
  RelIncl m linearly_orders  support b by PRE_POLY:82;
  then
A6: rng  S = support b by PRE_POLY:def 2;
A7: dom xc = m = dom xr & m = dom b by PARTFUN1:def 2;
A8: dom (xc*S) = dom S & dom (xr*S) = dom S &
  dom (xr*S) = dom S by A6,A7,RELAT_1:27;
  for i st 1<=i <= len S holds yC.i = yR.i
  proof
    let i such that
A9:1<=i<=len S;
A10:i in NAT & (b * S)/.i in NAT by ORDINAL1:def 12;
    reconsider r=(xr * S)/.i as Real;
A11: i in dom S & i in dom yR & i in dom yC by A9,FINSEQ_3:25,A2,A4;
A12: (xc * S)/.i = (xc * S).i = xc.(S.i) &
    (xr * S)/.i = (xr * S).i = xr.(S.i)
    by PARTFUN1:def 6,A9,FINSEQ_3:25,A8,FUNCT_1:12;
    then
A13: ex x being Real st x = (xc * S)/.i &
    (power F_Complex).((xc * S)/.i,(b * S)/.i) =
    x |^ (b * S)/.i by A1,UNIROOTS:43;
    thus yR.i = yR/.i by A11,PARTFUN1:def 6
    .=power(FR).((xr * S)/.i,(b * S)/.i) by A10,A9,A4,A5
    .=(power FC).((xc * S)/.i,(b * S)/.i) by NIVEN:7,A13,A12,A1
    .= yC/.i by A10,A9,A2,A3
    .=yC.i by A11,PARTFUN1:def 6;
  end;
  hence thesis by Th67,A4,A2,FINSEQ_1:14;
end;
