 reserve a for non empty set;
 reserve b, x, o for object;

theorem Th2:
    for a,b be non empty set holds ex c be non empty set st a /\ c = {} &
    (ex f be Function st f is one-to-one & dom f = b & rng f = c )
    proof
      let a,b be non empty set;
      consider d be object such that
A1:   for x be set holds not [x,d] in a by Th1;
      set C = [:b,{d}:];
      consider f be Function such that
A2:   f is one-to-one & dom f = b & rng f = C
        by WELLORD2:def 4,CARD_1:69;
A3:   a /\ C = {}
      proof
        assume a /\ C <> {}; then
        consider x be object such that
A5:     x in a /\ C by XBOOLE_0:def 1;
        x in C by A5,XBOOLE_0:def 4; then
        consider y,z be object such that
A6:     y in b & z in {d} & x = [y,z] by ZFMISC_1:84;
A7:     x = [y,d] by A6,TARSKI:def 1;
        reconsider y as set by A6;
        not [y,d] in a by A1;
        hence contradiction by A7, A5,XBOOLE_0:def 4;
      end;
      take C;
      thus thesis by A3,A2;
    end;
