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 Th31:
 for x,y being object holds
  x in Ch"{y} implies Intersection(F,Ch|(dom Ch\{x}),y)/\F.x=
  Intersection(F,Ch,y)
proof let x,y be object;
  set d=dom Ch\{x};
  set Chd=Ch|d;
  set I1=Intersection(F,Ch,y);
  set I2=Intersection(F,Chd,y);
  assume x in Ch"{y};
  then
A1: I1 c= F.x by Th30;
A2: I2/\F.x c= I1
  proof
    let z be object such that
A3: z in I2/\F.x;
    now
      let x1 such that
A4:   x1 in dom Ch and
A5:   Ch.x1=y;
      per cases by A4,XBOOLE_0:def 5;
      suppose
A6:     x1 in d;
A7:     z in I2 by A3,XBOOLE_0:def 4;
A8:     dom Ch/\d= dom Chd & dom Ch/\d =d by RELAT_1:61,XBOOLE_1:28;
        then Chd.x1=y by A5,A6,FUNCT_1:47;
        hence z in F.x1 by A6,A8,A7,Def2;
      end;
      suppose
        x1 in {x};
        then x1=x by TARSKI:def 1;
        hence z in F.x1 by A3,XBOOLE_0:def 4;
      end;
    end;
    hence thesis by A3,Def2;
  end;
  I1 c= I2 by Th26;
  then I1 c= I2/\F.x by A1,XBOOLE_1:19;
  hence thesis by A2;
end;
