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 Th57:
  F is_subformula_of G & G is_subformula_of H implies F is_subformula_of H
proof
  assume that
A1: F is_subformula_of G and
A2: G is_subformula_of H;
  now
    assume F <> G;
    then
A3: F is_proper_subformula_of G by A1;
    now
      assume G <> H;
      then G is_proper_subformula_of H by A2;
      then F is_proper_subformula_of H by A3,Th56;
      hence thesis;
    end;
    hence thesis by A1;
  end;
  hence thesis by A2;
end;
