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 Th84:
  dom @RestrictSub(x,p,Sub) misses dom ((@Sub)|RSub1(p)) \/ dom ((
  @Sub)|RSub2(p,Sub))
proof
  set X = {y : y in still_not-bound_in p & y is Element of dom Sub & y <> x &
  y <> Sub.y};
A1: dom ((@Sub)|RSub2(p,Sub)) = dom @Sub /\ RSub2(p,Sub) by RELAT_1:61;
  RestrictSub(x,p,Sub) = Sub|X by SUBSTUT1:def 6;
  then RestrictSub(x,p,Sub) = (@Sub|X) by SUBSTUT1:def 2;
  then dom @RestrictSub(x,p,Sub) = dom (@Sub|X) by SUBSTUT1:def 2;
  then
A2: dom @RestrictSub(x,p,Sub) = dom @Sub /\ X by RELAT_1:61;
A3: dom ((@Sub)|RSub1(p)) = dom @Sub /\ RSub1(p) by RELAT_1:61;
  now
    assume dom @RestrictSub(x,p,Sub) meets dom ((@Sub)|RSub1(p)) \/ dom ((@
    Sub)|RSub2(p,Sub));
    then consider b being object such that
A4: b in dom @RestrictSub(x,p,Sub) and
A5: b in dom ((@Sub)|RSub1(p)) \/ dom ((@Sub)|RSub2(p,Sub)) by XBOOLE_0:3;
    b in X by A2,A4,XBOOLE_0:def 4;
    then
A6: ex y st b = y & y in still_not-bound_in p & y is Element of dom Sub &
    y <> x & y <> Sub.y;
A7: now
      assume b in dom ((@Sub)|RSub2(p,Sub));
      then b in RSub2(p,Sub) by A1,XBOOLE_0:def 4;
      then ex y1 st y1 = b & y1 in still_not-bound_in p & y1 = (@ Sub).y1 by
Def10;
      hence contradiction by A6,SUBSTUT1:def 2;
    end;
    now
      assume b in dom ((@Sub)|RSub1(p));
      then b in RSub1(p) by A3,XBOOLE_0:def 4;
      then ex y1 st y1 = b & not y1 in still_not-bound_in p by Def9;
      hence contradiction by A6;
    end;
    hence contradiction by A5,A7,XBOOLE_0:def 3;
  end;
  hence thesis;
end;
