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;
reserve k,n,m for Nat;
reserve l for Element of omega;

theorem Th60:
  for f being Function holds card f = card dom f
proof
  let f be Function;
  dom f,f are_equipotent
  proof
    deffunc F(object) = [$1,f.$1];
    consider g being Function such that
A1: dom g = dom f and
A2: for x being object st x in dom f holds g.x = F(x) from FUNCT_1:sch 3;
    take g;
    thus g is one-to-one
    proof
      let x,y be object;
      assume that
A3:   x in dom g and
A4:   y in dom g;
      assume g.x = g.y;
      then [x,f.x] = g.y by A1,A2,A3
        .= [y,f.y] by A1,A2,A4;
      hence thesis by XTUPLE_0:1;
    end;
    thus dom g = dom f by A1;
    thus rng g c= f
    proof
      let i be object;
      assume i in rng g;
      then consider x being object such that
A5:   x in dom g and
A6:   g.x = i by FUNCT_1:def 3;
      g.x = [x,f.x] by A1,A2,A5;
      hence thesis by A1,A5,A6,FUNCT_1:1;
    end;
    let x,y be object;
    assume
A7: [x,y] in f;
    then
A8: x in dom f by FUNCT_1:1;
    y = f.x by A7,FUNCT_1:1;
    then g.x = [x,y] by A2,A7,FUNCT_1:1;
    hence thesis by A1,A8,FUNCT_1:def 3;
  end;
  hence thesis by Th4;
end;
