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

theorem
  R is transitive iff for x,y,z st [x,y] in R & [y,z] in R holds [x,z] in R
proof
  hereby
    assume
A1: R is transitive;
    let x,y,z;
    assume that
A2: [x,y] in R and
A3: [y,z] in R;
A4: z in field R by A3,RELAT_1:15;
    x in field R & y in field R by A2,RELAT_1:15;
    hence [x,z] in R by A1,A2,A3,A4,Def8;
  end;
  assume for x,y,z st [x,y] in R & [y,z] in R holds [x,z] in R;
  then x in field R & y in field R & z in field R & [x,y] in R & [y,z] in R
  implies [x,z] in R;
  hence R is_transitive_in field R;
end;
