reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;

theorem Th5:
  p is atomic implies ex k,P,ll st p = P!ll
proof
  assume p is atomic;
  then consider
  k being Nat, P being (QC-pred_symbol of k,A), ll being
  QC-variable_list of k,A such that
A1: p = P!ll by QC_LANG1:def 18;
  reconsider k as Element of NAT by ORDINAL1:def 12;
  reconsider ll as QC-variable_list of k,A;
A2: { ll.m where m is Nat
     : 1 <= m & m <= len ll & ll.m in fixed_QC-variables(A) } = {} by A1,
CQC_LANG:7;
  { ll.i  where i is Nat:
   1 <= i & i <= len ll & ll.i in free_QC-variables(A) } = {} by A1,
CQC_LANG:7;
  then ll is CQC-variable_list of k,A by A2,CQC_LANG:5;
  hence thesis by A1;
end;
