
theorem Th9:
  for T1,T2 being DecoratedTree, x being set holds T1 is binary &
  T2 is binary iff x-tree (T1,T2) is binary
proof
  let T1,T2 be DecoratedTree, x be set;
  hereby
    assume T1 is binary & T2 is binary;
    then dom T1 is binary & dom T2 is binary;
    then
A1: tree(dom T1, dom T2) is binary by Th8;
    dom (x-tree (T1, T2)) = tree(dom T1, dom T2) by TREES_4:14;
    hence x-tree (T1,T2) is binary by A1;
  end;
  assume x-tree (T1,T2) is binary;
  then
A2: dom (x-tree (T1,T2)) is binary;
  dom (x-tree (T1, T2)) = tree(dom T1, dom T2) by TREES_4:14;
  then dom T1 is binary & dom T2 is binary by A2,Th8;
  hence thesis;
end;
