reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;

theorem
  SFX <> {} & SFY <> {} implies meet SFX \/ meet SFY c= meet UNION (SFX, SFY)
proof
  set y = the Element of SFX;
  set z = the Element of SFY;
  assume SFX <> {} & SFY <> {};
  then
A1: y \/ z in UNION(SFX,SFY) by Def4;
  let x be object;
  assume x in meet SFX \/ meet SFY;
  then
A2: x in meet SFX or x in meet SFY by XBOOLE_0:def 3;
  for Z st Z in UNION(SFX,SFY) holds x in Z
  proof
    let Z;
    assume Z in UNION(SFX,SFY);
    then consider X,Y such that
A3: X in SFX & Y in SFY and
A4: Z = X \/ Y by Def4;
    x in X or x in Y by A2,A3,Def1;
    hence thesis by A4,XBOOLE_0:def 3;
  end;
  hence thesis by A1,Def1;
end;
