reserve Y for non empty set,
  a, b for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  A, B for a_partition of Y;

theorem
  for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G={
  A,B} & A<>B holds CompF(A,G) = B
proof
  let G be Subset of PARTITIONS(Y);
  let A,B be a_partition of Y;
  assume that
A1: G={A,B} and
A2: A<>B;
A3: '/\' {B} c= B
  proof
    let x be object;
    assume x in '/\' {B};
    then consider h being Function, F being Subset-Family of Y such that
A4: dom h={B} and
A5: rng h = F and
A6: for d being set st d in {B} holds h.d in d and
A7: x=Intersect F and
    x<>{} by BVFUNC_2:def 1;
    rng h = {h.B} by A4,FUNCT_1:4;
    then
A8: x=meet ({h.B}) by A5,A7,SETFAM_1:def 9;
    B in {B} by TARSKI:def 1;
    then h.B in B by A6;
    hence thesis by A8,SETFAM_1:10;
  end;
A9: B c= '/\' {B}
  proof
    let x be object;
    set h0 = B .--> x;
A10: dom h0 = {B};
    assume
A11: x in B;
A12: for d being set st d in {B} holds h0.d in d
    proof
      let d be set;
      assume d in {B};
      then d=B by TARSKI:def 1;
      hence thesis by A11,FUNCOP_1:72;
    end;
A13: rng h0 = {x} by FUNCOP_1:8;
A14: x is set by TARSKI:1;
    rng h0 c= bool Y
    proof
      let m be object;
      assume m in rng h0;
      then m=x by A13,TARSKI:def 1;
      hence thesis by A11;
    end;
    then reconsider F0 = rng h0 as Subset-Family of Y;
    meet F0 = Intersect F0 by A13,SETFAM_1:def 9;
    then
A15: x=Intersect F0 by A13,SETFAM_1:10,A14;
    x<>{} by A11,EQREL_1:def 4;
    hence thesis by A10,A12,A15,BVFUNC_2:def 1;
  end;
  A in {A} by TARSKI:def 1;
  then
A16: {A} \ {A}={} by ZFMISC_1:60;
  G \ {A}={A} \/ {B} \ {A} by A1,ENUMSET1:1
    .= ({A} \ {A}) \/ ({B} \ {A}) by XBOOLE_1:42
    .= ({A} \ {A}) \/ {B} by A2,ZFMISC_1:14;
  then CompF(A,G)='/\' {B} by A16,BVFUNC_2:def 7;
  hence thesis by A3,A9;
end;
