reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;
reserve S for Sequence;

theorem Th33:
  X,Y are_equipotent & x in X & y in Y implies X \ { x },Y \ { y }
  are_equipotent
proof
  given f such that
A1: f is one-to-one and
A2: dom f = X and
A3: rng f = Y;
A4: X \ { x },f.:(X \ { x }) are_equipotent by A1,A2,Th32;
  assume that
A5: x in X and
A6: y in Y;
  f.x in Y by A2,A3,A5,FUNCT_1:def 3;
  then
A7: Y \ { f.x },Y \ { y } are_equipotent by A6,Th31;
  f.:(X \ { x }) = f.:X \ Im(f,x) by A1,FUNCT_1:64
    .= Y \ Im(f,x) by A2,A3,RELAT_1:113
    .= Y \ { f.x } by A2,A5,FUNCT_1:59;
  hence thesis by A4,A7,WELLORD2:15;
end;
