reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th35:
  for X be set, F be Subset-Family of X st F is finite for A be
  Subset of X st A is infinite & A c= union F ex Y be Subset of X st Y in F & Y
  /\ A is infinite
proof
  defpred P[object,object] means
    ex D2 being set st D2 = $2 & $1 in D2;
:: Choice !!!
  let X be set, F be Subset-Family of X such that
A1: F is finite;
  let A be Subset of X such that
A2: A is infinite and
A3: A c= union F;
  set I=INTERSECTION (F,{A});
  card [:F,{A}:] =card F by CARD_1:69;
  then card I c= card F by TOPGEN_4:25;
  then
A4: I is finite by A1;
A5: for x being object st x in A ex y being object st y in I & P[x,y]
  proof
    let x be object such that
A6: x in A;
    consider y such that
A7: x in y and
A8: y in F by A3,A6,TARSKI:def 4;
    take y/\A;
    A in {A} by TARSKI:def 1;
    hence y/\A in I by A8,SETFAM_1:def 5;
    take y/\A;
    thus thesis by A6,A7,XBOOLE_0:def 4;
  end;
  consider p be Function of A,I such that
A9: for x being object st x in A holds P[x,p.x] from FUNCT_2:sch 1(A5);
  consider x being object such that
A10: x in A by A2,XBOOLE_0:def 1;
  ex y being object st y in I & P[x,y] by A5,A10;
  then dom p=A by FUNCT_2:def 1;
  then consider t be object such that
A11: t in rng p and
A12: p"{t} is infinite by A2,A4,CARD_2:101;
  consider Y,Z be set such that
A13: Y in F and
A14: Z in {A} and
A15: t=Y/\Z by A11,SETFAM_1:def 5;
  reconsider Y as Subset of X by A13;
  take Y;
A16: Z = A by A14,TARSKI:def 1;
  p"{t} c= Y/\A
  proof
    let z be object such that
A17: z in p"{t};
    p.z in {t} by A17,FUNCT_1:def 7;
    then
A18:  p.z=t by TARSKI:def 1;
    P[z,p.z] by A9,A17;
    hence thesis by A15,A16,A18;
  end;
  hence thesis by A12,A13;
end;
