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 Th61: :: STIRL2_1:53
  rF is nonnegative-yielding implies
  (Sum rF=0 iff (len rF=0 or rF = len rF --> 0))
proof
assume A1:
  rF is nonnegative-yielding;
  hereby
    assume
A2: Sum rF=0;
    assume
A3:    len rF <>0;
  set L=len rF -->0;
    assume rF <> len rF -->0;
    then consider k  such that
A4: k in dom L & L.k <> rF.k by AFINSQ_1:8,FUNCOP_1:13;
    rF.k in rng rF by A4,FUNCT_1:def 3;
    then L.k = 0 & rF.k >=0 by A4,A1,FUNCOP_1:7,PARTFUN3:def 4;
    hence contradiction by A2,Th60,A1,A4,A3;
 end;
    assume len rF=0 or rF= len rF -->0 ;
    then Sum rF = 0 or Sum rF = len rF *0 by Th57,Def8,BINOP_2:1;
    hence thesis;
end;
