reserve i,j,k,n,m for Nat,
  x,y,z,y1,y2 for object, X,Y,D for set,
  p,q for XFinSequence;
reserve k1,k2 for Nat;
reserve D for non empty set,
  F,G for XFinSequence of D,
  b for BinOp of D,
  d,d1,d2 for Element of D;
reserve F for XFinSequence,
        rF,rF1,rF2 for real-valued XFinSequence,
        r for Real,
        cF,cF1,cF2 for complex-valued XFinSequence,
        c,c1,c2 for Complex;

theorem Th66: :: CARD_FIN:15
 for x,y being object
 for f be Function st f.y<>x holds (f|(dom f\{y}))"{x}=f"{x}
proof let x,y be object;
  let f be Function;
  set d=dom f\{y};
  assume
A1: f.y<>x;
A2: f"{x} c= (f|d)"{x}
  proof
A3: dom (f|d)=dom f/\d by RELAT_1:61;
    let x1 be object such that
A4: x1 in f"{x};
A5: f.x1 in {x} by A4,FUNCT_1:def 7;
    f.x1 in {x} by A4,FUNCT_1:def 7;
    then f.x1=x by TARSKI:def 1;
    then
A6: not x1 in {y} by A1,TARSKI:def 1;
    x1 in dom f by A4,FUNCT_1:def 7;
    then x1 in d by A6,XBOOLE_0:def 5;
    then
A7: x1 in dom (f|d) by A3,XBOOLE_0:def 4;
    then f.x1=(f|d).x1 by FUNCT_1:47;
    hence thesis by A7,A5,FUNCT_1:def 7;
  end;
  (f|d)"{x} c= f"{x}
  proof
    let x1 be object such that
A8: x1 in (f|d)"{x};
A9: (f|d).x1 in {x} by A8,FUNCT_1:def 7;
A10: x1 in dom (f|d) by A8,FUNCT_1:def 7;
    then dom (f|d)=dom f/\d & f.x1=(f|d).x1 by FUNCT_1:47,RELAT_1:61;
    hence thesis by A10,A9,FUNCT_1:def 7;
  end;
  hence thesis by A2;
end;
