
theorem Th1:
  for R1, S1 being set, R being transitive (Relation of R1),
  S being transitive Relation of S1 st R1 misses S1 holds R \/ S is transitive
proof
  let R1, S1 be set, R be transitive (Relation of R1),
  S be transitive Relation of S1 such that
A1: R1 misses S1;
  let p1, p2, p3 be object;
  set RS = R \/ S;
  set D = field RS;
  assume that
  p1 in D and p2 in D and
A2: p3 in D and
A3: [p1,p2] in RS and
A4: [p2,p3] in RS;
  per cases by A2,XBOOLE_0:def 3;
  suppose
A5: p3 in R1;
    then p2 in R1 by A1,A4,Lm1;
    then
A6: [p1,p2] in R by A1,A3,Lm1;
    [p2,p3] in R by A1,A4,A5,Lm1;
    then [p1,p3] in R by A6,RELAT_2:31;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
A7: p3 in S1;
    then p2 in S1 by A1,A4,Lm1;
    then
A8: [p1,p2] in S by A1,A3,Lm1;
    [p2,p3] in S by A1,A4,A7,Lm1;
    then [p1,p3] in S by A8,RELAT_2:31;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
