reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;
reserve x,y,z for set,
  S, X for non empty set,
  R for Relation of X;

theorem
  R is transitive implies R is_transitive_in S
proof
  assume R is transitive; then
A1: R is_transitive_in field R;
  let x,y,z be object;
  assume x in S & y in S & z in S;
  assume
A2: [x,y] in R; then
A3: x in field R by RELAT_1:15;
  assume
A4: [y,z] in R;
  then y in field R & z in field R by RELAT_1:15;
  hence thesis by A1,A2,A4,A3;
end;
