reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;

theorem Th7:
  for R being total transitive Relation of X
  for x,y being object
  holds [x,y] in R & [y,z] in R implies [x,z] in R
proof
  let R be total transitive Relation of X;
  let x,y be object;
  assume that
A1: [x,y] in R and
A2: [y,z] in R;
A3: z in X by A2,Lm1;
  field R = X by ORDERS_1:12;
  then
A4: R is_transitive_in X by RELAT_2:def 16;
  x in X & y in X by A1,Lm1;
  hence thesis by A1,A2,A3,A4;
end;
