theorem
  x is_maximal_in R & R is connected implies x is_superior_of R
proof
  assume that
A1: x is_maximal_in R and
A2: R is connected;
  thus
A3: x in field R by A1;
  let y;
  assume that
A4: y in field R and
A5: y <> x;
  R is_connected_in field R by A2;
  then [x,y] in R or [y,x] in R by A3,A4,A5;
  hence thesis by A1,A4;
end;
