reserve A for QC-alphabet;
reserve i,j,k for Nat;

theorem Th1:
  for x being set holds x in QC-variables(A) iff x in
  fixed_QC-variables(A) or x in free_QC-variables(A) or
  x in bound_QC-variables(A)
proof
  let x be set;
  thus x in QC-variables(A) implies x in fixed_QC-variables(A) or x in
  free_QC-variables(A) or x in bound_QC-variables(A)
  proof
    assume x in QC-variables(A);
    then x in [: {6}, NAT :] \/ [: {4,5}, QC-symbols(A) :] by QC_LANG1:def 3;
    then x in [: {6}, NAT :] or x in [: {4,5}, QC-symbols(A) :]
      by XBOOLE_0:def 3;
    then consider x1,x2 being object such that
A1: (x1 in {6} & x2 in NAT & x = [x1,x2]) or
    (x1 in {4,5} & x2 in QC-symbols(A) & x = [x1,x2]) by ZFMISC_1:def 2;
    (x1 in {6} & x2 in NAT & x = [x1,x2]) or
    ((x1 = 4 or x1 = 5) & x2 in QC-symbols(A) & x = [x1,x2])
    by A1,TARSKI:def 2;
   then ((x1 in {4} & x2 in QC-symbols(A)) or (x1 in {5} & x2 in QC-symbols(A))
         or (x1 in {6} & x2 in NAT)) & x = [x1,x2] by TARSKI:def 1;
    then x in [:{4},QC-symbols(A):] or x in [:{5}, QC-symbols(A):] or
         x in [:{6},NAT:] by ZFMISC_1:def 2;
    hence thesis by QC_LANG1:def 4,def 5,def 6;
  end;
  thus thesis;
end;
