reserve X for set, a,b,c,x,y,z for object;
reserve P,R for Relation;

theorem
  R is transitive iff R*R c= R
proof
  hereby
    assume R is transitive;
    then
A1: R is_transitive_in field R;
    now
      let a,b be object;
      assume [a,b] in R*R;
      then consider c being object such that
A2:   [a,c] in R and
A3:   [c,b] in R by RELAT_1:def 8;
A4:   c in field R by A2,RELAT_1:15;
      a in field R & b in field R by A2,A3,RELAT_1:15;
      hence [a,b] in R by A1,A2,A3,A4;
    end;
    hence R*R c= R;
  end;
  assume
A5: R*R c= R;
  let a,b,c;
  assume a in field R & b in field R & c in field R;
  assume [a,b] in R & [b,c] in R;
  then [a,c] in R*R by RELAT_1:def 8;
  hence thesis by A5;
end;
