reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;
reserve V, C for set;

theorem Th23:
  for p being set holds Card <*p*> = <*card p*>
proof
  let p be set;
  set Cp = <*card p*>;
A1: dom Cp = {1} by FINSEQ_1:2,38;
  now
    let x be object;
    assume x in dom Cp;
    then x = 1 by A1,TARSKI:def 1;
    hence Cp.x is Cardinal;
  end;
  then reconsider Cp as Cardinal-Function by CARD_3:def 1;
A2: dom <*p*> = {1} by FINSEQ_1:2,38;
  now
    let x be object;
    assume x in dom <*p*>;
    then
A3: x = 1 by A2,TARSKI:def 1;
    hence <*card p*>.x = card p
      .= card (<*p*>.x) by A3;
  end;
  then Card <*p*> = Cp by A1,A2,CARD_3:def 2;
  hence thesis;
end;
