reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th6:
  for X,Y be set holds UNION(bool X,bool Y)=bool(X\/Y)
proof
  let X,Y be set;
  thus UNION(bool X,bool Y) c= bool(X\/Y)
  proof
    let xy be object;
    assume xy in UNION(bool X,bool Y);
    then consider x,y be set such that
A1:   x in bool X & y in bool Y & xy = x\/y by SETFAM_1:def 4;
    x\/y c= X\/Y by A1,XBOOLE_1:13;
    hence thesis by A1;
  end;
  let x;
  reconsider A=x as set by TARSKI:1;
  assume x in bool(X\/Y);
  then
A2: A\X c= Y by XBOOLE_1:43;
  A/\X c= X by XBOOLE_1:17;
  then (A/\X)\/(A\X) in UNION(bool X,bool Y) by A2,SETFAM_1:def 4;
  hence thesis;
end;
