reserve D,D1,D2 for non empty set,
        d,d1,d2 for XFinSequence of D,
        n,k,i,j for Nat;

theorem Th6:
  for f be FinSequence, x,y be object st rng f c= {x,y} & x<>y
    holds card (f"{x}) + card (f"{y}) = len f
proof
  let f be FinSequence, A,B be object;
A1:{A}\/{B}={A,B} by ENUMSET1:1;
  assume that
A2:   rng f c= {A,B}
    and
A3:   A<>B;
  f"rng f c= f"{A,B} by A2,RELAT_1:143;
  then
A4: dom f = f"{A,B} by RELAT_1:132,RELAT_1:134
         .= (f"{A})\/(f"{B}) by RELAT_1:140,A1;
  f"{A} misses f"{B}
  proof
    assume f"{A} meets f"{B};
    then consider x be object such that
A5: x in f"{A} and
A6: x in f"{B} by XBOOLE_0:3;
A7: f.x in {A} by A5,FUNCT_1:def 7;
A8: f.x in {B} by A6,FUNCT_1:def 7;
    f.x = A by A7,TARSKI:def 1;
    hence thesis by A8,TARSKI:def 1,A3;
  end;
  hence card (f"{A}) + card (f"{B})= card dom f by A4,CARD_2:40
    .= card Seg len f by FINSEQ_1:def 3
    .= len f by FINSEQ_1:57;
end;
