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

theorem
  for D be non empty set, F,G be PartFunc of D,REAL, X be set st dom(F|X
  ) is finite & dom(F|X) = dom(G|X) holds Sum(F-G,X) = Sum(F,X) - Sum (G,X)
proof
  let D be non empty set, F,G be PartFunc of D,REAL, X be set;
  assume
A1: dom(F|X) is finite & dom(F|X) = dom(G|X);
  dom(((-1)(#)G)|X) = dom((-1)(#)G) /\ X by RELAT_1:61
    .= dom G /\ X by VALUED_1:def 5
    .= dom(G|X) by RELAT_1:61;
  hence Sum(F-G,X) = Sum(F,X) + Sum((-1)(#)G,X) by A1,Th78
    .= Sum(F,X) +(-1)* Sum(G,X) by A1,Th77
    .= Sum(F,X) - Sum(G,X);
end;
