reserve X,Y for set,
  x,x1,x2,y,y1,y2,z for set,
  f,g,h for Function;
reserve M for non empty set;
reserve D for non empty set;
reserve P for Relation;
reserve O for Order of X;
reserve R,P for Relation,
  X,X1,X2,Y,Z,x,y,z,u for set,
  g,h for Function,
  O for Order of X,
  D for non empty set,
  d,d1,d2 for Element of D,
  A1,A2,B for Ordinal,
  L,L1,L2 for Sequence;

theorem
  x is_inferior_of R iff x is_superior_of R~
proof
A1: field R = field(R~) by RELAT_1:21;
  thus x is_inferior_of R implies x is_superior_of R~
  proof
    assume that
A2: x in field R and
A3: for y st y in field R & y <> x holds [x,y] 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;
    [x,y] in R by A1,A3,A4,A5;
    hence thesis by RELAT_1:def 7;
  end;
  assume that
A6: x in field(R~) and
A7: for y st y in field(R~) & y <> x holds [y,x] 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;
  [y,x] in R~ by A1,A7,A8,A9;
  hence thesis by RELAT_1:def 7;
end;
