reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;
reserve Un for FamilySequence of T,
  r,r1,r2 for Real,
  n for Element of NAT;

theorem
  for ADD being BinOp of Funcs(the carrier of T,REAL) st (for f1,f2
  being RealMap of T holds ADD.(f1,f2)=f1+f2) for F being FinSequence of Funcs(
the carrier of T,REAL) st for n st 0 <> n & n <= len F holds F.n is continuous
  RealMap of T holds ADD "**" F is continuous RealMap of T
proof
  let ADD be BinOp of Funcs(the carrier of T,REAL) such that
A1: for f1,f2 being RealMap of T holds ADD.(f1,f2)=f1+f2;
  set Fu=Funcs(the carrier of T,REAL);
  let F be FinSequence of Funcs(the carrier of T,REAL) such that
A2: for n st 0 <> n & n <= len F holds F.n is continuous RealMap of T;
  reconsider ADDF = ADD "**" F as RealMap of T by FUNCT_2:66;
  now
    per cases;
    suppose
A3:   len F=0;
A4:   ADD is having_a_unity by A1,Th23;
      then ex x be Element of Fu st x is_a_unity_wrt ADD by SETWISEO:def 2;
      then
A5:   the_unity_wrt ADD is_a_unity_wrt ADD by BINOP_1:def 8;
      ADDF = the_unity_wrt ADD by A3,A4,FINSOP_1:def 1;
      hence thesis by A1,A5,Th24;
    end;
    suppose
A6:   len F<>0;
A7:   len F >=1
      proof
        assume len F < 1;
        then len F <1+0;
        hence thesis by A6,NAT_1:13;
      end;
      then consider f be sequence of Fu such that
A8:   f.1 = F.1 and
A9:   for n being Nat st 0 <> n & n < len F holds f.(n+1) = ADD.(f.n,F.(n+1))
       and
A10:  ADD "**" F = f.(len F) by FINSOP_1:1;
      defpred Con[Nat] means $1<=len F implies f.$1 is continuous RealMap of T;
A11:  for n be Nat st n>=1 holds Con[n] implies Con[n+1]
      proof
        let n be Nat such that
A12:    n>=1 and
A13:    Con[n];
        assume
A14:    n+1<=len F;
        reconsider n as Element of NAT by ORDINAL1:def 12;
A15:    n+0<n+1 by XREAL_1:8;
        then n<len F by A14,XXREAL_0:2;
        then
A16:    f.(n+1)=ADD.(f.n,F.(n+1)) by A9,A12;
        1+0<=n+1 by A12,XREAL_1:8;
        then (n+1) in Seg len F by A14,FINSEQ_1:1;
        then n+1 in dom F by FINSEQ_1:def 3;
        then F.(n+1) in rng F by FUNCT_1:def 3;
        then reconsider fn=f.n,Fn1=F.(n+1) as RealMap of T by FUNCT_2:66;
        Fn1 is continuous by A2,A14;
        then fn+Fn1 is continuous by A13,A14,A15,Th22,XXREAL_0:2;
        hence thesis by A1,A16;
      end;
A17:  Con[1] by A2,A8;
      for n be Nat st n>=1 holds Con[n] from NAT_1:sch 8(A17,A11);
      hence thesis by A7,A10;
    end;
  end;
  hence thesis;
end;
