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 Th64: :: CARD_FIN:44
   n in dom cF implies Sum (cF|n) + cF.n = Sum (cF|(n+1))
proof
  assume
A1:  n in dom cF;
    reconsider cF as XFinSequence of COMPLEX;
 addcomplex.(addcomplex "**" cF|n, cF.n) = addcomplex "**" cF|(n+1)
      by Th42,A1;
  hence thesis by BINOP_2:def 3;
end;
