reserve A for QC-alphabet;
reserve n,k,m for Nat;
reserve F,G,G9,H,H9 for Element of QC-WFF(A);
reserve t, t9, t99 for Element of dom tree_of_subformulae(F);
reserve x for set;

theorem Th18:
  t <> t9 & (tree_of_subformulae(F)).t = (tree_of_subformulae(F)).
  t9 implies not t,t9 are_c=-comparable
proof
  assume that
A1: t <> t9 and
A2: (tree_of_subformulae(F)).t = (tree_of_subformulae(F)).t9;
  assume
A3: t,t9 are_c=-comparable;
  per cases by A3;
  suppose
    t is_a_prefix_of t9;
    then t is_a_proper_prefix_of t9 by A1;
    hence contradiction by A2,Th16;
  end;
  suppose
    t9 is_a_prefix_of t;
    then t9 is_a_proper_prefix_of t by A1;
    hence contradiction by A2,Th16;
  end;
end;
