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;

theorem Th35:
  {x1,x2}=Ch"{y} implies Intersection(F,Ch,y)=F.x1 /\ F.x2
proof
  assume
A1: {x1,x2}=Ch"{y};
  per cases;
  suppose
A2: x1=x2;
    then Ch"{y}={x1} by A1,ENUMSET1:29;
    hence thesis by A2,Th34;
  end;
  suppose
A3: x1<>x2;
    Ch"{y}/\(dom Ch\{x1})=(Ch"{y}/\dom Ch)\{x1} & Ch"{y}/\dom Ch={x1,x2}
    by A1,RELAT_1:132,XBOOLE_1:28,49;
    then Ch"{y}/\(dom Ch\{x1})={x2} by A3,ZFMISC_1:17;
    then
A4: (Ch|(dom Ch\{x1}))"{y}={x2} by FUNCT_1:70;
    x1 in Ch"{y} by A1,TARSKI:def 2;
    then Intersection(F,Ch|(dom Ch\{x1}),y)/\F.x1=Intersection(F,Ch,y) by Th31;
    hence thesis by A4,Th34;
  end;
end;
