theorem Th21:
  v1 in C & v2 in C implies v1 in ProperPrefixes v2 or v2 is_a_prefix_of v1
proof
  assume
A1: v1 in C & v2 in C;
  then reconsider p = v1, q = v2 as Element of W;
  assume
A2: not v1 in ProperPrefixes v2;
A3: p,q are_c=-comparable by A1,Def3;
A4: not v1 is_a_proper_prefix_of v2 by A2,TREES_1:12;
 v1 is_a_prefix_of v2 or v2 is_a_prefix_of v1 by A3;
  hence thesis by A4;
end;
