reserve x,y,z,a,b,c,X,X1,X2,Y,Z for set,
  W,W1,W2 for Tree,
  w,w9 for Element of W,
  f for Function,
  D,D9 for non empty set,
  i,k,k1,k2,l,m,n for Nat,
  v,v1,v2 for FinSequence,
  p,q,r,r1,r2 for FinSequence of NAT;
reserve C for Chain of W,
  B for Branch of W;
reserve T,T1,T2 for DecoratedTree;

theorem Th36:
  (for x st x in D9 holds x is DecoratedTree of D) & D9 is c=-linear implies
  union D9 is DecoratedTree of D
proof
  assume that
A1: for x st x in D9 holds x is DecoratedTree of D and
A2: D9 is c=-linear;
 for x st x in D9 holds x is DecoratedTree by A1;
  then reconsider T = union D9 as DecoratedTree by A2,Th35;
 rng T c= D
  proof
    let x be object;
    assume x in rng T;
    then consider y being object such that
A3: y in dom T & x = T.y by FUNCT_1:def 3;
 [y,x] in T by A3,FUNCT_1:1;
    then consider X such that
A4: [y,x] in X and
A5: X in D9 by TARSKI:def 4;
    reconsider X as DecoratedTree of D by A1,A5;
 y in dom X & x = X.y by A4,FUNCT_1:1;
then A6: x in rng X by FUNCT_1:def 3;
    thus thesis by A6;
  end;
  hence thesis by RELAT_1:def 19;
end;
