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;
reserve a for Element of A;

theorem Th82:
  dom ((@Sub)|RSub1(p)) misses dom ((@Sub)|RSub2(p,Sub))
proof
  now
    assume dom ((@Sub)|RSub1(p)) meets dom ((@Sub)|RSub2(p,Sub));
    then consider a being object such that
A1: a in dom ((@Sub)|RSub1(p)) /\ dom ((@Sub)|RSub2(p,Sub)) by XBOOLE_0:4;
    dom ((@Sub)|RSub1(p)) = dom (@Sub) /\ RSub1(p) & dom ((@Sub)|RSub2(p,
    Sub)) = (dom (@Sub) /\ RSub2(p,Sub)) by RELAT_1:61;
    then a in (dom (@Sub) /\ (dom (@Sub) /\ RSub1(p))) /\ RSub2(p,Sub) by A1,
XBOOLE_1:16;
    then a in dom (@Sub) /\ dom (@Sub) /\ RSub1(p) /\ RSub2(p,Sub) by
XBOOLE_1:16;
    then a in dom (@Sub) /\ (RSub1(p) /\ RSub2(p,Sub)) by XBOOLE_1:16;
    then
A2: a in RSub1(p) /\ RSub2(p,Sub) by XBOOLE_0:def 4;
    then a in RSub2(p,Sub) by XBOOLE_0:def 4;
    then
A3: ex b being bound_QC-variable of Al st b = a & b in still_not-bound_in p & b
    = (@Sub).b by Def10;
    a in RSub1(p) by A2,XBOOLE_0:def 4;
    then ex b being bound_QC-variable of Al st b = a & not b in
     still_not-bound_in p by Def9;
    hence contradiction by A3;
  end;
  hence thesis;
end;
