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 Th27:
  still_not-bound_in (X \/ Y) = still_not-bound_in X \/ still_not-bound_in Y
proof
  thus still_not-bound_in (X \/ Y) c=
  still_not-bound_in X \/ still_not-bound_in Y
  proof
    set A = {still_not-bound_in p : p in X \/ Y};
    let b be object;
    assume b in still_not-bound_in (X \/ Y);
    then consider a such that
A1: b in a and
A2: a in A by TARSKI:def 4;
    consider p such that
A3: a = still_not-bound_in p and
A4: p in X \/ Y by A2;
A5: now
      assume p in X;
      then a in {still_not-bound_in q : q in X} by A3;
      then
A6:   b in union {still_not-bound_in q : q in X} by A1,TARSKI:def 4;
      still_not-bound_in X c= still_not-bound_in X \/ still_not-bound_in Y
      by XBOOLE_1:7;
      hence thesis by A6;
    end;
    now
      assume p in Y;
      then a in {still_not-bound_in q : q in Y} by A3;
      then
A7:   b in union {still_not-bound_in q : q in Y} by A1,TARSKI:def 4;
      still_not-bound_in Y c= still_not-bound_in X \/ still_not-bound_in Y
      by XBOOLE_1:7;
      hence thesis by A7;
    end;
    hence thesis by A4,A5,XBOOLE_0:def 3;
  end;
  thus still_not-bound_in X \/ still_not-bound_in Y c=
  still_not-bound_in (X \/ Y)
  proof
    let b be object such that
A8: b in still_not-bound_in X \/ still_not-bound_in Y;
A9: now
      assume b in still_not-bound_in X;
      then consider a such that
A10:  b in a & a in {still_not-bound_in p : p in X} by TARSKI:def 4;
A11:  ex p st ( a = still_not-bound_in p)&( p in X) by A10;
      X c= X \/ Y by XBOOLE_1:7;
      then a in {still_not-bound_in q : q in X \/ Y} by A11;
      hence thesis by A10,TARSKI:def 4;
    end;
    now
      assume b in still_not-bound_in Y;
      then consider a such that
A12:  b in a & a in {still_not-bound_in p : p in Y} by TARSKI:def 4;
A13:  ex p st ( a = still_not-bound_in p)&( p in Y) by A12;
      Y c= X \/ Y by XBOOLE_1:7;
      then a in {still_not-bound_in q : q in X \/ Y} by A13;
      hence thesis by A12,TARSKI:def 4;
    end;
    hence thesis by A8,A9,XBOOLE_0:def 3;
  end;
end;
