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 Th85:
  [S,x] is quantifiable implies @(CQCSub_All([S,x],xSQ))`2 = @
RestrictSub(x,All(x,S`1),xSQ) +* (@xSQ)|RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x
  ,S`1),xSQ)
proof
  set S1 = CQCSub_All([S,x],xSQ);
A1: (@xSQ)|RSub2(All(x,S`1),xSQ) c= @xSQ by RELAT_1:59;
  dom ((@xSQ)|RSub1(All(x,S`1))) misses dom ((@xSQ)|RSub2(All(x,S`1),xSQ))
  by Th82;
  then
A2: ((@xSQ)|RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x,S`1),xSQ)) = ((@xSQ)|
  RSub1(All(x,S`1)) \/ (@xSQ)|RSub2(All(x,S`1),xSQ)) by FUNCT_4:31;
  assume
A3: [S,x] is quantifiable;
  then S1 = Sub_All([S,x],xSQ) by Def5;
  then
A4: @S1`2 = @xSQ by A3,Th26;
A5: @RestrictSub(x,All(x,S`1),xSQ) = @xSQ \ ((@xSQ)|RSub1(All(x,S`1)) +* (@
  xSQ)|RSub2(All(x,S`1),xSQ)) by Th83;
  then reconsider
  F = @xSQ \ ((@xSQ)|RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x,S`1),
  xSQ)) as PartFunc of bound_QC-variables(Al),bound_QC-variables(Al);
  dom F misses (dom ((@xSQ)|RSub1(All(x,S`1))) \/ dom ((@xSQ)|RSub2(All(x
  ,S`1),xSQ))) by A5,Th84;
  then
A6: dom F misses dom ((@xSQ)|RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x,S`1),
  xSQ)) by FUNCT_4:def 1;
  ((@xSQ)|RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x,S`1),xSQ)) \/ F = ((@xSQ
  )| RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x,S`1),xSQ)) \/ @xSQ & (@xSQ)|RSub1(
  All( x,S`1)) c= @xSQ by RELAT_1:59,XBOOLE_1:39;
  then
  ((@xSQ)|RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x,S`1),xSQ)) \/ F = @ xSQ
  by A2,A1,XBOOLE_1:8,12;
  then F +* ((@xSQ)|RSub1(All(x,S`1)) +* (@xSQ)|RSub2(All(x,S`1),xSQ)) = @xSQ
  by A6,FUNCT_4:31;
  hence thesis by A4,A5,FUNCT_4:14;
end;
