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

theorem Th63:
  for D be non empty set, F be PartFunc of D,REAL, X be set st dom
  (F|X) is finite holds FinS(F,dom(F|X)) = FinS(F,X)
proof
  let D be non empty set, F be PartFunc of D,REAL, X be set;
A1: F|(dom(F|X)) = F|(dom F /\ X) by RELAT_1:61
    .= (F|dom F)|X by RELAT_1:71
    .= F|X by RELAT_1:68;
  assume
A2: dom(F|X) is finite;
  then FinS(F,X), F|X are_fiberwise_equipotent by Def13;
  hence thesis by A2,A1,Def13;
end;
