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 Th35:
  (for x st x in D holds x is DecoratedTree) & D is c=-linear
  implies union D is DecoratedTree
proof
  assume that
A1: for x st x in D holds x is DecoratedTree and
A2: D is c=-linear;
 for x holds x in D implies x is Function by A1;
  then reconsider T = union D as Function by A2,Th34;
  defpred X[object,object] means ex T1 st $1 = T1 & dom T1 = $2;
A3: for x being object st x in D ex y being object st X[x,y]
  proof
    let x be object;
    assume x in D;
    then reconsider T1 = x as DecoratedTree by A1;
 dom T1 = dom T1;
    hence thesis;
  end;
  consider f such that
A4: dom f = D &
for x being object st x in D holds X[x,f.x] from CLASSES1:sch 1(A3);
  reconsider E = rng f as non empty set by A4,RELAT_1:42;
 now
    let x;
    assume x in E;
    then consider y being object such that
A5: y in dom f & x = f.y by FUNCT_1:def 3;
 ex T1 st y = T1 & dom T1 = x by A4,A5;
    hence x is Tree;
  end;
then A6: union E is Tree by Th33;
 dom T = union E
  proof
    thus dom T c= union E
    proof
      let x be object;
      assume x in dom T;
      then consider y being object such that
A7:  [x,y] in T by XTUPLE_0:def 12;
      consider X such that
A8:  [x,y] in X and
A9:  X in D by A7,TARSKI:def 4;
      consider T1 such that
A10:  X = T1 and
A11:  dom T1 = f.X by A4,A9;
A12:  dom T1 in rng f by A4,A9,A11,FUNCT_1:def 3;
A13:  x in dom T1 by A8,A10,XTUPLE_0:def 12;
  dom T1 c= union E by A12,ZFMISC_1:74;
      hence thesis by A13;
    end;
    let x be object;
    assume x in union E;
    then consider X such that
A14: x in X and
A15: X in E by TARSKI:def 4;
    consider y being object such that
A16: y in dom f and
A17: X = f.y by A15,FUNCT_1:def 3;
    consider T1 such that
A18: y = T1 and
A19: dom T1 = X by A4,A16,A17;
 [x,T1.x] in T1 by A14,A19,FUNCT_1:1;
then  [x,T1.x] in union D by A4,A16,A18,TARSKI:def 4;
    hence thesis by XTUPLE_0:def 12;
  end;
  hence thesis by A6,Def8;
end;
