
theorem Th16:
  for X be set for Y be finite Subset-Family of X for A,B be set
  st A in Components(Y) & B in Components(Y) & A <> B holds A misses B
proof
  let X be set;
  let Y be finite Subset-Family of X;
  let A,B be set;
  assume that
A1: A in Components(Y) and
A2: B in Components(Y) and
A3: A <> B;
  assume A /\ B <> {};
  then consider z be object such that
A4: z in A /\ B by XBOOLE_0:def 1;
A5: z in B by A4,XBOOLE_0:def 4;
A6: z in A by A4,XBOOLE_0:def 4;
  consider p be FinSequence of bool X such that
  len p = card Y and
  rng p = Y and
A7: Components(Y) = { Intersect (rng MergeSequence(p,q)) where q is
  FinSequence of BOOLEAN : len q = len p } by Def2;
  consider q1 be FinSequence of BOOLEAN such that
A8: A = Intersect (rng MergeSequence(p,q1)) and
  len q1 = len p by A1,A7;
  consider q2 be FinSequence of BOOLEAN such that
A9: B = Intersect (rng MergeSequence(p,q2)) and
  len q2 = len p by A2,A7;
A10: len MergeSequence(p,q1) = len p by Def1;
  then
A11: dom MergeSequence(p,q1) = Seg len p by FINSEQ_1:def 3;
A12: now
    let i be Nat;
    assume
A13: i in dom MergeSequence(p,q1);
    then
A14: i in dom p by A11,FINSEQ_1:def 3;
    MergeSequence(p,q1).i in rng MergeSequence(p,q1) by A13,FUNCT_1:def 3;
    then
A15: z in MergeSequence(p,q1).i by A8,A6,SETFAM_1:43;
    i in dom MergeSequence(p,q2) by A14,Th1;
    then MergeSequence(p,q2).i in rng MergeSequence(p,q2) by FUNCT_1:def 3;
    then
A16: z in MergeSequence(p,q2).i by A9,A5,SETFAM_1:43;
      per cases by XBOOLEAN:def 3;
      suppose
        q1.i = TRUE;
        then
A17:    MergeSequence(p,q1).i = p.i by Th2;
        q2.i = TRUE
        proof
          assume not q2.i = TRUE;
          then MergeSequence(p,q2).i = X\p.i by A14,Th3,XBOOLEAN:def 3;
          hence contradiction by A15,A16,A17,XBOOLE_0:def 5;
        end;
        hence MergeSequence(p,q1).i = MergeSequence(p,q2).i by A17,Th2;
      end;
      suppose q1.i = FALSE;
        then
A18:    MergeSequence(p,q1).i = X\p.i by A14,Th3;
        q2.i = FALSE
        proof
          assume not q2.i = FALSE;
          then q2.i = TRUE by XBOOLEAN:def 3;
          then MergeSequence(p,q2).i = p.i by Th2;
          hence contradiction by A15,A16,A18,XBOOLE_0:def 5;
        end;
        hence MergeSequence(p,q1).i = MergeSequence(p,q2).i by A14,A18,Th3;
      end;
  end;
  len MergeSequence(p,q2) = len p by Def1;
  hence contradiction by A3,A8,A9,A10,A12,FINSEQ_2:9;
end;
