reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th26:
  for A being Subset of Vars holds
  varcl {[varcl A, j]} = (varcl A) \/ {[varcl A, j]}
proof
  let A be Subset of Vars;
A1: {[varcl A, j]} c= (varcl A) \/ {[varcl A, j]} by XBOOLE_1:7;
A2: varcl A c= (varcl A) \/ {[varcl A, j]} by XBOOLE_1:7;
  now
    let x,y;
    assume [x,y] in (varcl A) \/ {[varcl A, j]};
    then [x,y] in varcl A or [x,y] in {[varcl A, j]} by XBOOLE_0:def 3;
    then [x,y] in varcl A or [x,y] = [varcl A, j] by TARSKI:def 1;
    then x c= varcl A or x = varcl A by Def1,XTUPLE_0:1;
    hence x c= (varcl A) \/ {[varcl A, j]} by A2;
  end;
  hence varcl {[varcl A, j]} c= (varcl A) \/ {[varcl A, j]} by A1,Def1;
A3: {[varcl A, j]} c= varcl {[varcl A, j]} by Def1;
  [varcl A, j] in {[varcl A, j]} by TARSKI:def 1;
  then varcl A c= varcl {[varcl A, j]} by A3,Def1;
  hence thesis by A3,XBOOLE_1:8;
end;
