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

theorem
  for R being FinSequence of REAL, F being FinSequence of COMPLEX st R =
  F & len R >= 1 holds addreal $$ R = addcomplex $$ F
proof
  let R be FinSequence of REAL,F be FinSequence of COMPLEX;
  assume that
A1: R = F and
A2: len R >= 1;
  consider f22 be sequence of  REAL such that
A3: f22.1 = R.1 and
A4: for n be Nat st 0 <> n & n < len R holds f22.(n + 1) =
  addreal.(f22.n,R.(n + 1)) and
A5: addreal $$ R = f22.len R by A2,FINSOP_1:1;
  defpred P[set,set] means $2= f22.$1;
A6: for k be Nat st k in Seg len R ex x being Element of REAL st P[k,x]
    proof let k be Nat;
     f22.k in REAL by XREAL_0:def 1;
     hence thesis;
    end;
  ex f2 being FinSequence of REAL st dom f2 = Seg len R & for k be Nat st
  k in Seg len R holds P[k,f2.k] from FINSEQ_1:sch 5(A6);
  then consider f2 being FinSequence of REAL such that
A7: dom f2 = Seg len R and
A8: for k be Nat st k in Seg len R holds P[k,f2.k];
  consider f9 being sequence of COMPLEX such that
A9: for n being Nat st 1<= n & n<=len FR2FC f2 holds f9.n=(FR2FC f2).n by Th19;
A10: len f2 = len R by A7,FINSEQ_1:def 3;
  then
A11: (FR2FC f2).(len F)=f9.(len F) by A1,A2,A9;
A12: for n st 0 <> n & n < len R holds f2.(n + 1) = addreal.(f2.n,R.(n + 1))
  proof
    let n;
    assume that
A13: 0 <> n and
A14: n < len R;
A15: n+1<=len R by A14,NAT_1:13;
A16: 0+1<=n by A13,NAT_1:13;
    then
A17: n in Seg len R by A14,FINSEQ_1:1;
    1<=n+1 by A16,NAT_1:13;
    then (n+1) in Seg len R by A15,FINSEQ_1:1;
    then f2.(n + 1) = f22.(n + 1) by A8
      .= addreal.(f22.n,R.(n + 1)) by A4,A13,A14
      .= addreal.(f2.n,R.(n + 1)) by A8,A17;
    hence thesis;
  end;
A18: for n be Nat st 0 <> n & n < len F holds f9.(n + 1) =
  addcomplex.(f9.n,F.(n + 1))
  proof
    let n be Nat;
    assume that
A19: 0 <> n and
A20: n < len F;
A21: 0+1<=n by A19,NAT_1:13;
    n<=len FR2FC f2 by A1,A7,A20,FINSEQ_1:def 3;
    then f9.n=(FR2FC f2).n by A9,A21;
    then
A22: addcomplex.(f9.n,F.(n + 1)) =addreal.(f2.n,R.(n + 1))
        by A1,COMPLSP2:44;
    n+1<=len (FR2FC f2) by A1,A10,A20,NAT_1:13;
    then f9.(n+1)=(FR2FC f2).(n+1) by A9,NAT_1:11;
    hence thesis by A1,A12,A19,A20,A22;
  end;
  set d = addreal $$ R;
A23: 1 in Seg len R by A2,FINSEQ_1:1;
A24: f9.1=(FR2FC f2).1 by A2,A10,A9;
  len R in Seg len R by A2,FINSEQ_1:1;
  then (FR2FC f2).len F = d by A1,A5,A8;
  hence thesis by A1,A2,A3,A8,A23,A24,A18,A11,FINSOP_1:2;
end;
