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 Th45:
  still_not-bound_in p c= Bound_Vars(p) & still_not-bound_in q c=
  Bound_Vars(q) implies still_not-bound_in (p '&' q) c= Bound_Vars(p '&' q)
proof
A1: still_not-bound_in (p '&' q) = still_not-bound_in p \/
  still_not-bound_in q by QC_LANG3:10;
  p '&' q is conjunctive by QC_LANG1:def 20;
  then Bound_Vars(p '&' q) = Bound_Vars(the_left_argument_of (p '&' q)) \/
  Bound_Vars(the_right_argument_of (p '&' q)) by SUBSTUT1:5;
  then
  Bound_Vars(p '&' q) = Bound_Vars(p) \/ Bound_Vars(the_right_argument_of
  (p '&' q)) by QC_LANG2:4;
  then
A2: Bound_Vars(p '&' q) = Bound_Vars(p) \/ Bound_Vars(q) by QC_LANG2:4;
  assume still_not-bound_in p c= Bound_Vars(p) & still_not-bound_in q c=
  Bound_Vars(q );
  hence thesis by A2,A1,XBOOLE_1:13;
end;
