reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;
reserve F,Ch for Function;
reserve Fy for finite-yielding Function;

theorem Th48:
  dom Intersect(F,dom F-->X9)=dom F & for x st x in dom F holds
  Intersect(F,dom F-->X9).x = F.x /\ X9
proof
A1: dom F/\dom(dom F-->X9)=dom F;
  hence dom F=dom Intersect(F,dom F-->X9) by YELLOW20:def 2;
  let x;
  assume
A2: x in dom F;
  then Intersect(F,dom F-->X9).x=F.x/\(dom F-->X9).x by A1,YELLOW20:def 2;
  hence thesis by A2,FUNCOP_1:7;
end;
