reserve Al for QC-alphabet;
reserve a,b,b1 for object,
  i,j,k,n for Nat,
  p,q,r,s for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  P for QC-pred_symbol of k,Al,
  l,ll for CQC-variable_list of k,Al,
  Sub,Sub1 for CQC_Substitution of Al,
  S,S1,S2 for Element of CQC-Sub-WFF(Al),
  P1,P2 for Element of QC-pred_symbols(Al);

theorem
  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,Al),
l being QC-variable_list of k,Al such that
A1: p = P!l by QC_LANG1:def 18;
A2: { l.j : 1 <= j & j <= len l & l.j in fixed_QC-variables(Al) } = {} by A1,
CQC_LANG:7;
  { l.i : 1 <= i & i <= len l & l.i in free_QC-variables(Al) } = {} by A1,
CQC_LANG:7;
  then reconsider l as CQC-variable_list of k,Al by A2,CQC_LANG:5;
  take k,P,l;
  thus thesis by A1;
end;
