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

theorem Th28:
   for F be FinSequence of the carrier of Polynom-Ring INT.Ring,
   x be Element of INT.Ring, x1 be Element of F_Real st x = x1 holds
   eval(^F,x1) = eval(F,x)
   proof
     set PRI = Polynom-Ring INT.Ring;
     set PRR = Polynom-Ring F_Real;
     let F be FinSequence of the carrier of Polynom-Ring INT.Ring,
         x be Element of INT.Ring, x1 be Element of F_Real;
     assume
A1:  x = x1;
A2:  dom eval(^F,x1) = dom ^F by Def8 .= dom F by Def7;
A3:  dom eval(^F,x1) = dom ^F by Def8;
     for i be Nat st i in dom eval(^F,x1) holds
     (eval(^F,x1)).i = (eval(F,x)).i
     proof
       let i be Nat;
       assume
A4:    i in dom eval(^F,x1); then
       F.i in rng F by A2,FUNCT_1:3; then
       reconsider Fi = F.i as Element of the carrier of PRI;
A5:    (^F)/.i = (^F).i by A3,A4,PARTFUN1:def 6
       .= ^(F/.i) by A4,A2,Def7 .= ^Fi by A4,A2,PARTFUN1:def 6;
       reconsider rFi = (^F)/.i as Element of the carrier of PRR;
A7:    F.i = F/.i by A4,A2,PARTFUN1:def 6;
A8:    (eval(^F,x1)).i = eval(~rFi,x1) by A4,A3,Def8 .= eval(rFi,x1);
       (eval(F,x)).i = eval(~(F/.i),x) by A4,A2,Def8 .= eval(Fi,x) by A7;
       hence thesis by A8,A1,A5,FIELD_4:27;
     end;
     hence thesis by A2,Def8;
   end;
