reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem
  for D be non empty set, F be PartFunc of D,REAL, X,Y be set st dom(F|(
X \/ Y)) is finite & dom(F|X) misses dom(F|Y) holds Sum(F, X \/ Y) = Sum(F,X) +
  Sum(F,Y)
proof
  let D be non empty set, F be PartFunc of D,REAL, X,Y be set;
  assume that
A1: dom(F|(X \/ Y)) is finite and
A2: dom(F|X) misses dom(F|Y);
A3: dom(F|(X \/ Y)) = dom F /\ (X \/ Y) by RELAT_1:61
    .= dom F /\ X \/ dom F /\ Y by XBOOLE_1:23
    .= dom(F|X) \/ dom F /\ Y by RELAT_1:61
    .= dom(F|X) \/ dom(F|Y) by RELAT_1:61;
  then dom(F|X) is finite by A1,FINSET_1:1,XBOOLE_1:7;
  then
A4: FinS(F,X) = FinS(F,dom(F| X)) by Th63;
  dom(F|Y) is finite by A1,A3,FINSET_1:1,XBOOLE_1:7;
  then
A5: FinS(F,Y) = FinS(F,dom(F|Y)) by Th63;
A6: dom(F| dom(F|(X \/ Y))) = dom F /\ dom(F|(X \/ Y)) by RELAT_1:61
    .= dom F /\ (dom F /\ (X \/ Y)) by RELAT_1:61
    .= dom F /\ dom F /\ (X \/ Y) by XBOOLE_1:16
    .= dom(F|(X \/ Y)) by RELAT_1:61;
  FinS(F,X \/ Y) = FinS(F,dom(F|(X \/ Y))) by A1,Th63;
  hence Sum(F, X \/ Y) = Sum (FinS(F,X) ^ FinS(F,Y)) by A1,A2,A3,A4,A5,A6,Th76,
RFINSEQ:9
    .= Sum(F,X) + Sum(F,Y) by RVSUM_1:75;
end;
