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 Th63:
  F is_proper_subformula_of G & G is_subformula_of H or F
  is_subformula_of G & G is_proper_subformula_of H or F is_subformula_of G & G
  is_immediate_constituent_of H or F is_immediate_constituent_of G & G
  is_subformula_of H or F is_proper_subformula_of G & G
  is_immediate_constituent_of H or F is_immediate_constituent_of G & G
  is_proper_subformula_of H implies F is_proper_subformula_of H
proof
  assume
A1: F is_proper_subformula_of G & G is_subformula_of H or F
  is_subformula_of G & G is_proper_subformula_of H or F is_subformula_of G & G
  is_immediate_constituent_of H or F is_immediate_constituent_of G & G
  is_subformula_of H or F is_proper_subformula_of G & G
  is_immediate_constituent_of H or F is_immediate_constituent_of G & G
  is_proper_subformula_of H;
  then F is_subformula_of G & G is_subformula_of H by Th52;
  hence F is_subformula_of H by Th57;
  thus thesis by A1,Th59,Th61;
end;
