
theorem Th3: :: U5s:
for A, B, C, D, E, X being set
 st X c= A or X c= B or X c= C or X c= D or X c= E
  holds X c= A \/ B \/ C \/ D \/ E
proof
 let A, B, C, D, E, X be set;
 assume A1: X c= A or X c= B or X c= C or X c= D or X c= E;
 per cases by A1;
  suppose X c= A;
    then X c= A \/ B by XBOOLE_1:10;
    then X c= A \/ B \/ C by XBOOLE_1:10;
    then X c= A \/ B \/ C \/ D by XBOOLE_1:10;
    hence X c= A \/ B \/ C \/ D \/ E by XBOOLE_1:10;
  end;
  suppose X c= B;
    then X c= A \/ B by XBOOLE_1:10;
    then X c= A \/ B \/ C by XBOOLE_1:10;
    then X c= A \/ B \/ C \/ D by XBOOLE_1:10;
    hence X c= A \/ B \/ C \/ D \/ E by XBOOLE_1:10;
  end;
  suppose X c= C;
    then X c= A \/ B \/ C by XBOOLE_1:10;
    then X c= A \/ B \/ C \/ D by XBOOLE_1:10;
    hence X c= A \/ B \/ C \/ D \/ E by XBOOLE_1:10;
  end;
  suppose X c= D;
    then X c= A \/ B \/ C \/ D by XBOOLE_1:10;
    hence X c= A \/ B \/ C \/ D \/ E by XBOOLE_1:10;
  end;
  suppose X c= E;
    hence X c= A \/ B \/ C \/ D \/ E by XBOOLE_1:10;
  end;
end;
