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

theorem Th66:
  for D be non empty set, d be Element of D, X be set,
      F be PartFunc of D,REAL st
      dom(F|X) is finite & d in dom(F|X) holds
    FinS(F,X\{d})^<*F.d*>, F|X are_fiberwise_equipotent
proof
  let D be non empty set, d be Element of D, X be set, F be PartFunc of D,REAL;
  set dx = dom(F|X);
  assume that
A1: dx is finite and
A2: d in dx;
  set Y = X \ {d}, dy = dom(F|Y);
A3: dy=dom F /\ Y by RELAT_1:61;
A4: dx=dom F /\ X by RELAT_1:61;
A5: dy = dx \ {d}
  proof
    thus dy c= dx \ {d}
    proof
      let y be object;
      assume
A6:   y in dy;
      then y in X \ {d} by A3,XBOOLE_0:def 4;
      then
A7:   not y in {d} by XBOOLE_0:def 5;
      y in dom F by A3,A6,XBOOLE_0:def 4;
      then y in dx by A3,A4,A6,XBOOLE_0:def 4;
      hence thesis by A7,XBOOLE_0:def 5;
    end;
    let y be object;
    assume
A8: y in dx \{d};
    then
A9: not y in {d} by XBOOLE_0:def 5;
A10: y in dx by A8,XBOOLE_0:def 5;
    then y in X by A4,XBOOLE_0:def 4;
    then
A11: y in Y by A9,XBOOLE_0:def 5;
    y in dom F by A4,A10,XBOOLE_0:def 4;
    hence thesis by A3,A11,XBOOLE_0:def 4;
  end;
  F|dx = F|(dom F /\ X) by RELAT_1:61
    .=(F|dom F)|X by RELAT_1:71
    .=F|X by RELAT_1:68;
  then FinS(F,dx\{d})^<*F.d*>, F|X are_fiberwise_equipotent by A1,A2,Th65;
  hence thesis by A1,A5,Th63;
end;
