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;
reserve x1,x2 for set,
  w for FinSequence of NAT;

theorem
  for X,Y being set for B being c=-linear Subset of PFuncs(X,Y)
  holds union B in PFuncs(X,Y)
proof
  let X,Y be set;
  let B be c=-linear Subset of PFuncs(X,Y);
  for x be set st x in B holds x is Function;
  then reconsider f = union B as Function by Th34;
  per cases;
  suppose
    B <> {};
    then reconsider D = B as non empty functional set;
A1: now
      let x be set;
      assume x in the set of all  dom g where g is Element of D;
      then consider g being Element of D such that
A2:   x = dom g;
      g in PFuncs(X,Y) by TARSKI:def 3;
      then ex f being Function st g = f & dom f c= X & rng f c= Y by
PARTFUN1:def 3;
      hence x c= X by A2;
    end;
A3: now
      let x be set;
      assume x in the set of all  rng g where g is Element of D;
      then consider g being Element of D such that
A4:   x = rng g;
      g in PFuncs(X,Y) by TARSKI:def 3;
      then ex f being Function st g = f & dom f c= X & rng f c= Y by
PARTFUN1:def 3;
      hence x c= Y by A4;
    end;
    rng f = union the set of all  rng g where g is Element of D
          by FUNCT_1:110;
    then
A5: rng f c= Y by A3,ZFMISC_1:76;
    dom f = union the set of all  dom g where g is Element of D
     by FUNCT_1:110;
    then dom f c= X by A1,ZFMISC_1:76;
    hence thesis by A5,PARTFUN1:def 3;
  end;
  suppose
A6: B = {};
    {} is PartFunc of X, Y by RELSET_1:12;
    hence thesis by A6,PARTFUN1:45,ZFMISC_1:2;
  end;
end;
