reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th19:
  for F being FinSequence of COMPLEX ex G being sequence of
  COMPLEX st for n being Nat st 1<= n & n<=len F holds G.n=F.n
proof
  let F be FinSequence of COMPLEX;
  defpred P[object,object] means
    ($1 in Seg len F implies $2=F.$1) & (not $1 in Seg len F implies $2=0);
A1: for x being object st x in NAT
   ex y being object st y in COMPLEX & P[x,y]
  proof
    let x be object;
    assume x in NAT;
    per cases;
    suppose
 A2:   x in Seg len F;
      take F.x;
       F.x in COMPLEX by XCMPLX_0:def 2;
      hence thesis by A2;
    end;
    suppose
A3:    not x in Seg len F;
      take 0;
       0 in COMPLEX by XCMPLX_0:def 2;
      hence thesis by A3;
    end;
  end;
  ex G1 being sequence of COMPLEX st for x being object st x in NAT holds
  P[x,G1.x] from FUNCT_2:sch 1(A1);
  then consider G2 being sequence of COMPLEX such that
A4: for x being object st x in NAT holds (x in Seg len F implies G2.x=F.x)
  & (not x in Seg len F implies G2.x=0);
  for n being Nat st 1<= n & n<=len F holds G2.n=F.n
  proof
    let n be Nat;
    assume that
A5: 1<= n and
A6: n<=len F;
    n in Seg len F by A5,A6,FINSEQ_1:1;
    hence thesis by A4;
  end;
  hence thesis;
end;
