
theorem
  for X be set for Y,Z be finite Subset-Family of X st Z c= Y holds
  Components(Y) is_finer_than Components(Z)
proof
  let X be set;
  let Y,Z be finite Subset-Family of X;
  assume
A1: Z c= Y;
  now
    let V be set;
    consider p be FinSequence of bool X such that
A2: len p = card Y and
A3: rng p = Y and
A4: Components(Y) = { Intersect (rng MergeSequence(p,q)) where q is
    FinSequence of BOOLEAN : len q = len p } by Def2;
    consider p1 be FinSequence of bool X such that
    len p1 = card Z and
A5: rng p1 = Z and
A6: Components(Z) = { Intersect (rng MergeSequence(p1,q)) where q is
    FinSequence of BOOLEAN : len q = len p1 } by Def2;
A7: p1 is FinSequence of rng p by A1,A3,A5,FINSEQ_1:def 4;
    assume V in Components(Y);
    then consider q be FinSequence of BOOLEAN such that
A8: V = Intersect (rng MergeSequence(p,q)) and
A9: len q = len p by A4;
    dom p = dom q by A9,FINSEQ_3:29;
    then
A10: q is Function of dom p,BOOLEAN by FINSEQ_2:26;
A11: p is one-to-one by A2,A3,FINSEQ_4:62;
    then
A12: rng p1 c= dom (p") by A1,A3,A5,FUNCT_1:33;
    rng (p") = dom p by A11,FUNCT_1:33
      .= dom q by A9,FINSEQ_3:29;
    then
A13: rng (p"*p1) c= dom q by RELAT_1:26;
    p is Function of dom p,rng p by FUNCT_2:1;
    then p" is Function of rng p,dom p by A11,FUNCT_2:25;
    then p"*p1 is FinSequence of dom p by A7,FINSEQ_2:32;
    then q*(p"*p1) is FinSequence of BOOLEAN by A10,FINSEQ_2:32;
    then reconsider q1 = q*p"*p1 as FinSequence of BOOLEAN by RELAT_1:36;
    reconsider W = Intersect (rng MergeSequence(p1,q1)) as set;
    take W;
    dom q1 = dom (q*(p"*p1)) by RELAT_1:36
      .= dom (p"*p1) by A13,RELAT_1:27
      .= dom p1 by A12,RELAT_1:27;
    then len q1 = len p1 by FINSEQ_3:29;
    hence W in Components(Z) by A6;
    rng MergeSequence(p1,q1) c= rng MergeSequence(p,q)
    proof
      let z be object;
      assume z in rng MergeSequence(p1,q1);
      then consider i be Nat such that
A14:  i in dom MergeSequence(p1,q1) and
A15:  MergeSequence(p1,q1).i = z by FINSEQ_2:10;
A16:  i in dom p1 by A14,Th1;
      then
A17:  i in dom (p"*p1) by A12,RELAT_1:27;
      then (p"*p1).i in rng (p"*p1) by FUNCT_1:def 3;
      then
A18:  (p"*p1).i in rng (p") by FUNCT_1:14;
      then
A19:  (p"*p1).i in dom p by A11,FUNCT_1:33;
      then reconsider j = (p"*p1).i as Element of NAT;
A20:  q.j = (q*(p"*p1)).i by A17,FUNCT_1:13
        .= q1.i by RELAT_1:36;
A21:  p1 is Function of dom p1,rng p by A1,A3,A5,FUNCT_2:2;
A22:  j in dom p by A11,A18,FUNCT_1:33;
A23:  now
        per cases by XBOOLEAN:def 3;
        suppose
A24:      q.j = TRUE;
          hence MergeSequence(p,q).j = p.j by Th2
            .= (p*(p"*p1)).i by A17,FUNCT_1:13
            .= (p*p"*p1).i by RELAT_1:36
            .= ((id rng p)*p1).i by A11,FUNCT_1:39
            .= p1.i by A21,FUNCT_2:17
            .= z by A15,A20,A24,Th2;
        end;
        suppose
A25:     q.j = FALSE;
          hence MergeSequence(p,q).j = X\p.j by A22,Th3
            .= X\(p*(p"*p1)).i by A17,FUNCT_1:13
            .= X\(p*p"*p1).i by RELAT_1:36
            .= X\((id rng p)*p1).i by A11,FUNCT_1:39
            .= X\p1.i by A21,FUNCT_2:17
            .= z by A15,A16,A20,A25,Th3;
        end;
      end;
      j in dom MergeSequence(p,q) by A19,Th1;
      hence thesis by A23,FUNCT_1:def 3;
    end;
    hence V c= W by A8,SETFAM_1:44;
  end;
  hence thesis by SETFAM_1:def 2;
end;
