reserve A for QC-alphabet;
reserve sq for FinSequence,
  x,y,z for bound_QC-variable of A,
  p,q,p1,p2,q1 for Element of QC-WFF(A);
reserve s,t for bound_QC-variable of A;
reserve F,G,H,H1 for Element of QC-WFF(A);
reserve x,y,z for bound_QC-variable of A,
  k,n,m for Nat,
  P for ( QC-pred_symbol of k, A),
  V for QC-variable_list of k, A;
reserve L,L9 for FinSequence;

theorem Th58:
  G is_subformula_of H & H is_subformula_of G implies G = H
proof
  assume that
A1: G is_subformula_of H and
A2: H is_subformula_of G;
  assume
A3: G <> H;
  then G is_proper_subformula_of H by A1;
  then
A4: len @G < len @H by Th54;
  H is_proper_subformula_of G by A2,A3;
  hence contradiction by A4,Th54;
end;
