reserve X,x for set;
reserve k,m,n for Element of NAT,
  p,q,r,s,r9,s9 for Element of HP-WFF,
  T1,T2 for Tree;
reserve T1,T2 for DecoratedTree;

theorem Th7:
  x-tree(T1,T2).<*0*> = T1.{} & x-tree(T1,T2).<*1*> = T2.{}
proof
A1: len<*T1,T2*> = 2 by FINSEQ_1:44;
  reconsider w = {} as Node of T1 by TREES_1:22;
A2: <*T1,T2*>.(0+1) = T1;
  thus x-tree(T1,T2).<*0*> = (x-tree<*T1,T2*>).<*0*> by TREES_4:def 6
    .= (x-tree<*T1,T2*>).(<*0*>^w) by FINSEQ_1:34
    .= T1.{} by A1,A2,TREES_4:12;
  reconsider w = {} as Node of T2 by TREES_1:22;
A3: <*T1,T2*>.(1+1) = T2;
  thus x-tree(T1,T2).<*1*> = (x-tree<*T1,T2*>).<*1*> by TREES_4:def 6
    .= (x-tree<*T1,T2*>).(<*1*>^w) by FINSEQ_1:34
    .= T2.{} by A1,A3,TREES_4:12;
end;
