
theorem LemCon:
  for X being object, R being Relation holds
    R is connected iff
      for x, y being object st x <> y & x in field R & y in field R
      holds [x,y] in R or [y,x] in R
  proof
    let X be object, R be Relation;
    thus R is connected implies
      for x, y being object st x <> y & x in field R & y in field R
      holds [x,y] in R or [y,x] in R by RELAT_2:def 6,def 14;
    assume
A1: for x, y being object st x <> y & x in field R & y in field R
      holds [x,y] in R or [y,x] in R;
    set X = field R;
    for x,y being object st x in X & y in X & x <> y
      holds [x,y] in R or [y,x] in R by A1;
    hence R is connected by RELAT_2:def 14,def 6;
  end;
