reserve Al for QC-alphabet;
reserve b,c,d for set,
  X,Y for Subset of CQC-WFF(Al),
  i,j,k,m,n for Nat,
  p,p1,q,r,s,s1 for Element of CQC-WFF(Al),
  x,x1,x2,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al, A,
  v for Element of Valuations_in(Al,A),
  f1,f2 for FinSequence of CQC-WFF(Al),
  CX,CY,CZ for Consistent Subset of CQC-WFF(Al),
  JH for Henkin_interpretation of CX,
  a for Element of A,
  t,u for QC-symbol of Al;
reserve L for PATH of q,p,
  F1,F3 for QC-formula of Al,
  a for set;
reserve C,D for Element of [:CQC-WFF(Al),bool bound_QC-variables(Al):];
reserve K,L for Subset of bound_QC-variables(Al);

theorem Th29:
  X c= Y implies still_not-bound_in X c= still_not-bound_in Y
proof
  set A = {still_not-bound_in p : p in X};
  assume
A1: X c= Y;
    let a be object;
    assume a in still_not-bound_in X;
    then consider b such that
A2: a in b and
A3: b in A by TARSKI:def 4;
    ex p st ( b = still_not-bound_in p)&( p in X) by A3;
    then b in {still_not-bound_in q : q in Y} by A1;
    hence a in still_not-bound_in Y by A2,TARSKI:def 4;
end;
