
theorem :: Exercise 4.29.ii
  for R being Relation st R is transitive holds R\~ is transitive
proof
  let R be Relation;
  assume R is transitive;
  then
A1: R is_transitive_in field R;
  now
    let x, y, z be object such that
    x in field (R\~) and y in field (R\~)
    and z in field (R\~) and
A2: [x,y] in R\~ and
A3: [y,z] in R\~;
A4: x in field R by A2,RELAT_1:15;
A5: y in field R by A2,RELAT_1:15;
A6: z in field R by A3,RELAT_1:15;
    then
A7: [x,z] in R by A1,A2,A3,A4,A5;
    not [x,y] in R~ by A2,XBOOLE_0:def 5;
    then not [y,x] in R by RELAT_1:def 7;
    then not [z,x] in R by A1,A3,A4,A5,A6;
    then not [x,z] in R~ by RELAT_1:def 7;
    hence [x,z] in R\~ by A7,XBOOLE_0:def 5;
  end;
  then R\~ is_transitive_in field (R\~);
  hence thesis;
end;
