reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;

theorem Th43:
  still_not-bound_in (P!ll) = Bound_Vars(P!ll)
proof
  set l1 = the_arguments_of (P!ll);
A1: P!ll is atomic by QC_LANG1:def 18;
  then consider
  n being Nat, P9 being (QC-pred_symbol of n,Al), ll9 being
  QC-variable_list of n,Al such that
A2: l1 = ll9 and
A3: P!ll = P9!ll9 by QC_LANG1:def 23;
  Bound_Vars(P!ll) = Bound_Vars(l1) by A1,SUBSTUT1:3;
  then
A4: Bound_Vars(P!ll) = { l1.i : 1 <= i & i <= len l1 & l1.i in
  bound_QC-variables(Al)} by SUBSTUT1:def 7;
  still_not-bound_in (P!ll) = still_not-bound_in ll by QC_LANG3:5;
  then
A5: still_not-bound_in (P!ll) = variables_in(ll,bound_QC-variables(Al)) by
QC_LANG3:2;
A6: (<*P9*>^ll9).1 = P9 & (<*P*>^ll).1 = P by FINSEQ_1:41;
  P!ll = <*P*>^ll & P9!ll9 = <*P9*>^ll9 by QC_LANG1:8;
  then ll9 = ll by A3,A6,FINSEQ_1:33;
  hence thesis by A4,A5,A2,QC_LANG3:def 1;
end;
