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 Th26:
  for X being set st X c= bound_QC-variables(Al) holds not x in X
  implies v.(x|a)|X = v|X
proof
  let X be set such that
A1: X c= bound_QC-variables(Al) and
A2: not x in X;
  set f2 = v|X;
  set f1 = v.(x|a)|X;
A3: dom f1 = dom f2 by A1,SUBLEMMA:63;
  now
    let b be object such that
A4: b in dom f1;
    x <> b by A2,A4;
    then
A5: v.(x|a).b = v.b by SUBLEMMA:48;
    v.(x|a).b = f1.b by A4,FUNCT_1:47;
    hence f1.b = f2.b by A3,A4,A5,FUNCT_1:47;
  end;
  hence thesis by A3,FUNCT_1:2;
end;
