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 be set st dom(F|X)
  is finite & (for d be Element of D st d in dom(F|X) holds F.d>=0) holds FinS(
  max+ F, X) = FinS(F, X)
proof
  let D be non empty set, F be PartFunc of D,REAL, X be set;
  assume that
A1: dom(F|X) is finite and
A2: for d be Element of D st d in dom(F|X) holds F.d>=0;
  now
    let d be Element of D;
    assume
A3: d in dom(F|X);
    then F.d>=0 by A2;
    hence (F|X).d>=0 by A3,FUNCT_1:47;
  end;
  then
A4: (F|X) = max+ (F|X) by Th46
    .= (max+ F)|X by Th44;
  hence FinS(F,X) = FinS((max+ F)|X,X) by A1,Th64
    .= FinS(max+ F,X) by A1,A4,Th64;
end;
