reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th8:
  for X being set, R being Relation of X st X c= field R
  holds R[*] is_transitive_in X
proof
  let X be set, R be Relation of X such that
A1: X c= field R;
  now
    let x,y,z be object;
    assume that
A2: x in X and
    y in X and
    z in X and
A3: [x,y] in R[*] & [y,z] in R[*];
    R reduces x,y & R reduces y,z by A3,REWRITE1:20;
    hence [x,z] in R[*] by A1,A2,Th6,REWRITE1:16;
  end;
  hence thesis by RELAT_2:def 8;
end;
