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 Th66:
  v|(still_not-bound_in p \ {x}) = w|(still_not-bound_in p \ {x})
  implies v.(x|a)|still_not-bound_in p = w.(x|a)|still_not-bound_in p
proof
A1: dom (w.(x|a)|still_not-bound_in p) = still_not-bound_in p by Th63;
  then
A2: dom (v.(x|a)|still_not-bound_in p) = dom (w.(x|a)|still_not-bound_in p)
  by Th63;
  assume
A3: v|(still_not-bound_in p \ {x}) = w|(still_not-bound_in p \ {x});
  for b being object
st b in dom (v.(x|a)|still_not-bound_in p) holds (v.(x|a)|
  still_not-bound_in p).b = (w.(x|a)|still_not-bound_in p).b
  proof
    let b being object such that
A4: b in dom (v.(x|a)|still_not-bound_in p);
A5: (v.(x|a)|still_not-bound_in p).b = v.(x|a).b & (w.(x|a)|
    still_not-bound_in p ).b = w.(x|a).b by A2,A4,FUNCT_1:47;
A6: now
      assume
A7:   b <> x;
      then
A8:   not b in {x} by TARSKI:def 1;
      b in still_not-bound_in p by A4,Th63;
      then
A9:   b in still_not-bound_in p \ {x} by A8,XBOOLE_0:def 5;
      then b in dom (w|(still_not-bound_in p \ {x})) by Th63;
      then
A10:  w|(still_not-bound_in p \ {x}).b = w.b by FUNCT_1:47;
A11:  v.(x|a).b = v.b & w.(x|a).b = w.b by A7,Th48;
      b in dom (v|(still_not-bound_in p \ {x})) by A9,Th63;
      hence thesis by A3,A5,A10,A11,FUNCT_1:47;
    end;
    now
A12:  w.(x|a)|(still_not-bound_in p).b = w.(x|a).b by A2,A4,FUNCT_1:47;
      assume
A13:  b = x;
      v.(x|a)|(still_not-bound_in p).b = v.(x|a).b by A4,FUNCT_1:47;
      then v.(x|a)|(still_not-bound_in p).b = a by A13,Th49;
      hence thesis by A13,A12,Th49;
    end;
    hence thesis by A6;
  end;
  hence thesis by A1,Th63,FUNCT_1:2;
end;
