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;

theorem Th9:
  card X c= card Y iff ex f st f is one-to-one & dom f = X & rng f c= Y
proof
  thus card X c= card Y implies ex f st f is one-to-one & dom f = X & rng f c=
  Y
  proof
    consider f such that
A1: f is one-to-one and
A2: dom f = X & rng f = card X by Def2,WELLORD2:def 4;
    assume
A3: card X c= card Y;
    consider g such that
A4: g is one-to-one and
A5: dom g = Y & rng g = card Y by Def2,WELLORD2:def 4;
    take h = g"*f;
    thus h is one-to-one by A1,A4;
    rng g = dom(g") & dom g = rng(g") by A4,FUNCT_1:33;
    hence thesis by A3,A2,A5,RELAT_1:26,27;
  end;
  given f such that
A6: f is one-to-one and
A7: dom f = X & rng f c= Y;
  consider g such that
A8: g is one-to-one and
A9: dom g = Y and
A10: rng g = card Y by Def2,WELLORD2:def 4;
A11: X,rng(g*f) are_equipotent
  proof
    take g*f;
    thus g*f is one-to-one by A6,A8;
    thus dom(g*f) = X by A7,A9,RELAT_1:27;
    thus thesis;
  end;
A12: rng(g*f),card rng(g*f) are_equipotent by Def2;
  card rng(g*f) c= card Y by A10,Th6,RELAT_1:26;
  hence thesis by A12,Def2,A11,WELLORD2:15;
end;
