reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem Th70:
  X meets Y \/ Z iff X meets Y or X meets Z
proof
  thus X meets Y \/ Z implies X meets Y or X meets Z
  proof
    assume X meets Y \/ Z;
    then consider x being object such that
A1: x in X & x in Y \/ Z by XBOOLE_0:3;
    x in X & x in Y or x in X & x in Z by A1,XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:3;
  end;
A2: X meets Z implies X meets Y \/ Z
  proof
    assume X meets Z;
    then consider x being object such that
A3: x in X and
A4: x in Z by XBOOLE_0:3;
    x in Y \/ Z by A4,XBOOLE_0:def 3;
    hence thesis by A3,XBOOLE_0:3;
  end;
  X meets Y implies X meets Y \/ Z
  proof
    assume X meets Y;
    then consider x being object such that
A5: x in X and
A6: x in Y by XBOOLE_0:3;
    x in Y \/ Z by A6,XBOOLE_0:def 3;
    hence thesis by A5,XBOOLE_0:3;
  end;
  hence thesis by A2;
end;
