reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;
reserve I,j for set;
reserve f,g for Function of I, NAT;
reserve J,K for finite Subset of I;

theorem Th26:
  J misses K implies Sum(f|(J \/ K)) = Sum(f|J) + Sum(f|K)
proof
  assume
A1: J misses K;
  per cases;
  suppose
A2: I is empty;
    set b = EmptyBag {};
A3: Sum b = 0 by UPROOTS:11;
    J = {} & f|J = {} by A2;
    hence Sum(f|(J \/ K)) = Sum(f|J) + Sum(f|K) by A3;
  end;
  suppose
    I is non empty;
    then reconsider I9 = I as non empty set;
    [:I9,NAT:] c= [:I9,REAL:] by ZFMISC_1:95,NUMBERS:19;
    then reconsider F = f as PartFunc of I9, REAL by XBOOLE_1:1;
A4: dom(F|(J \/ K)) is finite;
    thus Sum(f|(J \/ K)) = Sum(F,J \/ K) by Th25
      .= Sum(F,J) + Sum(F,K) by A1,A4,RFUNCT_3:83
      .= Sum(f|J) + Sum(F,K) by Th25
      .= Sum(f|J) + Sum(f|K) by Th25;
  end;
end;
