reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);
reserve fin,fin1 for FinSequence;
reserve PR,PR1 for FinSequence of [:set_of_CQC-WFF-seq(Al),Proof_Step_Kinds:];
reserve a for Element of A;

theorem Th28:
  not y in still_not-bound_in All(x,p) implies v.(y|a).(x|a)|
  still_not-bound_in p = v.(x|a)|still_not-bound_in p
proof
A1: v.(y|a).(x|a) = v+*((y|a)+*(x|a)) by FUNCT_4:14;
  assume
A2: not y in still_not-bound_in All(x,p);
  now
    assume
A3: x <> y;
    dom (x|a) = {x} & dom (y|a) = {y};
    then v.(y|a).(x|a) = v+*((x|a)+*(y|a)) by A1,A3,FUNCT_4:35,ZFMISC_1:11;
    then
A4: v.(y|a).(x|a) = v+*(x|a)+*(y|a) by FUNCT_4:14;
    not y in (still_not-bound_in p) \ {x} by A2,QC_LANG3:12;
    then not y in still_not-bound_in p by A3,ZFMISC_1:56;
    hence thesis by A4,Th26;
  end;
  hence thesis by A1;
end;
