theorem Th59:
  x is_minimal_in R iff x is_maximal_in R~
proof
A1: field R = field(R~) by RELAT_1:21;
  thus x is_minimal_in R implies x is_maximal_in R~
  proof
    assume that
A2: x in field R and
A3: not ex y st y in field R & y <> x & [y,x] in R;
    thus x in field(R~) by A2,RELAT_1:21;
    let y;
    assume that
A4: y in field(R~) and
A5: y <> x;
    not [y,x] in R by A1,A3,A4,A5;
    hence thesis by RELAT_1:def 7;
  end;
  assume that
A6: x in field(R~) and
A7: not ex y st y in field(R~) & y <> x & [x,y] in R~;
  thus x in field R by A6,RELAT_1:21;
  let y;
  assume that
A8: y in field R and
A9: y <> x;
  not [x,y] in R~ by A1,A7,A8,A9;
  hence thesis by RELAT_1:def 7;
end;
