reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th47:
   F is real-valued implies Sum F = addreal "**" F
proof
    assume A1:F is real-valued;
then  rng F c= REAL by VALUED_0:def 3;
then A2:F is REAL-valued by RELAT_1:def 19;
  rng F c= COMPLEX by A1,MEMBERED:1;
  then A3:F is COMPLEX-valued by RELAT_1:def 19;
  per cases by NAT_1:14;
    suppose A4:len F=0;
      hence addreal "**" F = 0 by Def8,A2,BINOP_2:2
                          .= Sum F by Def8,A3,A4,BINOP_2:1;
    end;
    suppose A5:len F>=1;
        A6: REAL = REAL /\ COMPLEX by MEMBERED:1,XBOOLE_1:28;
  now let x,y;assume x in REAL & y in REAL;
  then reconsider X=x,Y=y as Element of REAL;
   addreal.(x,y) = X+Y by BINOP_2:def 9;
   hence addreal.(x,y) =addcomplex.(x,y) & addreal.(x,y) in REAL
     by BINOP_2:def 3,XREAL_0:def 1;
 end;
hence thesis by Th46,A5,A6,A2;
    end;
end;
