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 Th69:
  [S,x] is quantifiable implies (v.((S_Bound(@CQCSub_All([S,x],xSQ
)))|a)). (NEx_Val(v,S,x,xSQ)+*(x|a))|still_not-bound_in S`1 = (v.(NEx_Val(v,S,x
  ,xSQ)+*(x|a)))|still_not-bound_in S`1
proof
  set S1 = CQCSub_All([S,x],xSQ);
  set z = S_Bound(@S1);
  set finSub = RestrictSub(x,All(x,S`1),xSQ);
  set V1 = (v.(z|a)).(NEx_Val(v,S,x,xSQ)+*(x|a));
  set V2 = v.(NEx_Val(v,S,x,xSQ)+*(x|a));
  set X = still_not-bound_in S`1;
A1: dom V1 = dom (v.(z|a)) \/ dom (NEx_Val(v,S,x,xSQ)+*(x|a)) by FUNCT_4:def 1;
  dom (v.(z|a)) = bound_QC-variables(Al) by Th58;
  then dom (v.(z|a)) = dom v by Th58;
  then
A2: dom V1 = dom V2 by A1,FUNCT_4:def 1;
  assume
A3: [S,x] is quantifiable;
A4: now
    assume not x in rng finSub;
    then
A5: z = x by A3,Th52;
    for b being object st b in dom V1 holds V1.b = V2.b
    proof
      let b being object such that
A6:   b in dom V1;
A7:   now
        assume
A8:     b <> z;
A9:     now
A10:      not b in dom (x|a) by A5,A8,TARSKI:def 1;
          assume not b in dom NEx_Val(v,S,x,xSQ);
          then not b in dom NEx_Val(v,S,x,xSQ) \/ dom (x|a) by A10,
XBOOLE_0:def 3;
          then
A11:      not b in dom (NEx_Val(v,S,x,xSQ)+*(x|a)) by FUNCT_4:def 1;
          reconsider x = b as bound_QC-variable of Al by A6;
          V1.b = (v.(z|a)).b by A11,FUNCT_4:11;
          then V1.b = v.x by A8,Th48;
          hence thesis by A11,FUNCT_4:11;
        end;
        now
          dom (NEx_Val(v,S,x,xSQ)+*(x|a)) = dom NEx_Val(v,S,x,xSQ) \/ dom
          (x|a) by FUNCT_4:def 1;
          then
A12:      dom NEx_Val(v,S,x,xSQ) c= dom (NEx_Val(v,S,x,xSQ)+*(x| a)) by
XBOOLE_1:7;
          assume
A13:      b in dom NEx_Val(v,S,x,xSQ);
          then V1.b = (NEx_Val(v,S,x,xSQ)+*(x|a)).b by A12,FUNCT_4:13;
          hence thesis by A13,A12,FUNCT_4:13;
        end;
        hence thesis by A9;
      end;
      now
        assume b = z;
        then b in {x} by A5,TARSKI:def 1;
        then
A14:    b in dom (x|a);
        dom (NEx_Val(v,S,x,xSQ)+*(x|a)) = dom NEx_Val(v,S,x,xSQ) \/ dom (
        x|a) by FUNCT_4:def 1;
        then
A15:    dom (x|a) c= dom (NEx_Val(v,S,x,xSQ)+*(x|a)) by XBOOLE_1:7;
        then V1.b = (NEx_Val(v,S,x,xSQ)+*(x|a)).b by A14,FUNCT_4:13;
        hence thesis by A14,A15,FUNCT_4:13;
      end;
      hence thesis by A7;
    end;
    hence thesis by A2,FUNCT_1:2;
  end;
  now
    assume
A16: x in rng finSub;
A17: dom (V1|X) = (dom (v.(z|a)) \/ dom (NEx_Val(v,S,x,xSQ)+*(x|a))) /\ X
    by A1,RELAT_1:61;
A18: for b being object st b in dom (V1|X) holds V1|X.b = V2|X.b
    proof
A19:  X c= Bound_Vars(S`1) by Th47;
      let b being object such that
A20:  b in dom (V1|X);
      b in X by A17,A20,XBOOLE_0:def 4;
      then
A21:  b <> z by A3,A16,A19,Th38;
A22:  V2|X = v|X +* (NEx_Val(v,S,x,xSQ)+*(x|a))|X by FUNCT_4:71;
A23:  V1|X = (v.(z|a))|X +* (NEx_Val(v,S,x,xSQ)+*(x|a))|X by FUNCT_4:71;
      then
A24:  dom (V1|X) = dom (v.(z|a)|X) \/ dom ((NEx_Val(v,S,x,xSQ)+*(x|a))|X)
      by FUNCT_4:def 1;
A25:  now
        assume
A26:    not b in dom ((NEx_Val(v,S,x,xSQ)+*(x|a))|X);
        then
A27:    b in dom (v.(z|a)|X) by A20,A24,XBOOLE_0:def 3;
        then b in dom (v.(z|a)) /\ X by RELAT_1:61;
        then
A28:    b in X by XBOOLE_0:def 4;
        b in bound_QC-variables(Al) by A20;
        then b in dom v by Th58;
        then
A29:    b in dom v /\ X by A28,XBOOLE_0:def 4;
        V1|X.b = (v.(z|a))|X.b by A23,A26,FUNCT_4:11;
        then V1|X.b = v.(z|a).b by A27,FUNCT_1:47;
        then
A30:    V1|X.b = v.b by A21,Th48;
        V2|X.b = v|X.b by A22,A26,FUNCT_4:11;
        hence thesis by A30,A29,FUNCT_1:48;
      end;
      now
        assume
A31:    b in dom ((NEx_Val(v,S,x,xSQ)+*(x|a))|X);
        then V1|X.b = (NEx_Val(v,S,x,xSQ)+*(x|a))|X.b by A23,FUNCT_4:13;
        hence thesis by A22,A31,FUNCT_4:13;
      end;
      hence thesis by A25;
    end;
    dom (V2|X) = dom (V1) /\ X by A2,RELAT_1:61;
    hence thesis by A18,FUNCT_1:2,RELAT_1:61;
  end;
  hence thesis by A4;
end;
