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
) is finite & Y c= X & (for d1,d2 be Element of D st d1 in dom(F|Y) & d2 in dom
  (F|(X\Y)) holds F.d1>=F.d2) holds FinS(F,X) = FinS(F,Y) ^ FinS(F,X \ Y)
proof
A1: for n holds P[n] from NAT_1:sch 2(Lm3,Lm4);
  let D be non empty set, F be PartFunc of D,REAL, X be set;
  let Y be set;
  assume that
A2: dom(F|X) is finite and
A3: Y c= X and
A4: for d1,d2 be Element of D st d1 in dom(F|Y) & d2 in dom(F|(X\Y))
  holds F.d1>=F.d2;
  F|Y c= F|X by A3,RELAT_1:75;
  then reconsider dFY = dom(F|Y) as finite set by A2,FINSET_1:1,RELAT_1:11;
  card dFY = card dFY;
  hence thesis by A1,A2,A3,A4;
end;
