reserve A for RelStr;
reserve X for non empty set;
reserve PX,PY,PZ,Y,a,b,c,x,y for set;
reserve S1,S2 for Subset of Y;

theorem
  for C being set st C is Strong_Classification of X holds InclPoset
  union C is Tree
proof
A1: X in {X} by TARSKI:def 1;
A2: X in {X} by TARSKI:def 1;
  let C being set such that
A3: C is Strong_Classification of X;
A4: C is Classification of X by A3,TAXONOM1:def 2;
  set B = union C;
A5: {X} in C by A3,TAXONOM1:def 2;
  then reconsider B9 = B as non empty set by A2,TARSKI:def 4;
  set R9 = RelIncl B;
  reconsider R = R9 as Relation of B;
  set D = RelStr(#B,R#);
  {X} in C by A3,TAXONOM1:def 2;
  then
A6: B <> {} by A1,TARSKI:def 4;
A7: now
    let x,y be Element of D;
    given z be Element of D such that
A8: z <= x and
A9: z <= y;
    reconsider z9 = z as Element of B9;
    reconsider z99 = z9 as Subset of X by A4,Th3;
    consider Z be set such that
A10: z9 in Z and
A11: Z in C by TARSKI:def 4;
    reconsider Z9 = Z as a_partition of X by A3,A11,PARTIT1:def 3;
    z99 in Z9 by A10;
    then z99 <> {} by EQREL_1:def 4;
    then consider a be object such that
A12: a in z by XBOOLE_0:def 1;
    [z,y] in R by A9,ORDERS_2:def 5;
    then
A13: z c= y by A6,WELLORD2:def 1;
    then
A14: a in y by A12;
A15: C is Classification of X by A3,TAXONOM1:def 2;
    reconsider x9 = x, y9 = y as Element of B9;
    consider S be set such that
A16: x9 in S and
A17: S in C by TARSKI:def 4;
    reconsider S9 = S as a_partition of X by A3,A17,PARTIT1:def 3;
    consider T be set such that
A18: y9 in T and
A19: T in C by TARSKI:def 4;
    reconsider T9 = T as a_partition of X by A3,A19,PARTIT1:def 3;
    [z,x] in R by A8,ORDERS_2:def 5;
    then
A20: z c= x by A6,WELLORD2:def 1;
    then
A21: a in x by A12;
    now
      per cases by A17,A19,A15,TAXONOM1:def 1;
      suppose
        S9 is_finer_than T9;
        then ex Y be set st Y in T9 & x9 c= Y by A16;
        hence x9 c= y9 or y9 c= x9 by A12,A21,A13,A18,Lm1;
      end;
      suppose
        T9 is_finer_than S9;
        then ex Y be set st Y in S9 & y9 c= Y by A18;
        hence x9 c= y9 or y9 c= x9 by A12,A20,A14,A16,Lm1;
      end;
    end;
    then [x9,y9] in R or [y9,x9] in R by WELLORD2:def 1;
    hence x <= y or y <= x by ORDERS_2:def 5;
  end;
A22: D is with_superior
  proof
    reconsider C9 = C as Strong_Classification of X by A3;
    reconsider s = X as Element of D by A5,A2,TARSKI:def 4;
    consider x be object such that
A23: x in SmallestPartition X by XBOOLE_0:def 1;
    SmallestPartition X in C9 by TAXONOM1:def 2;
    then reconsider x9 = x as Element of D by A23,TARSKI:def 4;
    take s;
A24: now
      let y be set such that
A25:  y in field R and
      y <> s;
A26:  y in dom R \/ rng R by A25,RELAT_1:def 6;
      per cases by A26,XBOOLE_0:def 3;
      suppose
        y in dom R;
        then reconsider y9 = y as Element of B9;
        y9 c= s by A4,Th3;
        hence [y,s] in R by WELLORD2:def 1;
      end;
      suppose
        y in rng R;
        then reconsider y9 = y as Element of B9;
        y9 c= s by A4,Th3;
        hence [y,s] in R by WELLORD2:def 1;
      end;
    end;
    [x9,s] in R by A6,A23,WELLORD2:def 1;
    then s in field R by RELAT_1:15;
    hence thesis by A24,ORDERS_1:def 14;
  end;
  RelStr(#B,R#) = InclPoset B by YELLOW_1:def 1;
  hence thesis by A22,A7,Def2;
end;
