reserve A,B,a,b,c,d,e,f,g,h for set;

theorem Th8:
  for R1,R2 being RelStr holds union_of(R1,R2) = union_of(R2,R1) &
  sum_of(R1,R2) = sum_of(R2,R1)
proof
  let R1,R2 be RelStr;
  set U1 = union_of(R1,R2), S1 = sum_of(R1,R2);
A1: the carrier of S1 = (the carrier of R2) \/ the carrier of R1 by
NECKLA_2:def 3;
A2: the InternalRel of S1 = (the InternalRel of R1) \/ (the InternalRel of
  R2 ) \/ [:the carrier of R1, the carrier of R2:] \/ [:the carrier of R2, the
  carrier of R1:] by NECKLA_2:def 3
    .= (the InternalRel of R2) \/ (the InternalRel of R1) \/ [:the carrier
  of R2, the carrier of R1:] \/ [:the carrier of R1, the carrier of R2:] by
XBOOLE_1:4;
  the carrier of U1 = (the carrier of R2) \/ the carrier of R1 & the
  InternalRel of U1 = (the InternalRel of R2) \/ the InternalRel of R1 by
NECKLA_2:def 2;
  hence thesis by A1,A2,NECKLA_2:def 2,def 3;
end;
