
theorem Th2:
  for L,S being RelStr holds (S is SubRelStr of L iff S opp is
  SubRelStr of L opp) & (S opp is SubRelStr of L iff S is SubRelStr of L opp)
proof
  let L,S be RelStr;
  thus S is SubRelStr of L implies S opp is SubRelStr of L opp
  proof
    assume that
A1: the carrier of S c= the carrier of L and
A2: the InternalRel of S c= the InternalRel of L;
    thus the carrier of S opp c= the carrier of L opp & the InternalRel of S
    opp c= the InternalRel of L opp by A1,A2,Th1;
  end;
  thus S opp is SubRelStr of L opp implies S is SubRelStr of L
  proof
    assume that
A3: the carrier of S opp c= the carrier of L opp and
A4: the InternalRel of S opp c= the InternalRel of L opp;
    thus the carrier of S c= the carrier of L & the InternalRel of S c= the
    InternalRel of L by A3,A4,Th1;
  end;
  thus S opp is SubRelStr of L implies S is SubRelStr of L opp
  proof
    assume that
A5: the carrier of S opp c= the carrier of L and
A6: the InternalRel of S opp c= the InternalRel of L;
    thus the carrier of S c= the carrier of L opp & the InternalRel of S c=
    the InternalRel of L opp by A5,A6,Th1;
  end;
  assume that
A7: the carrier of S c= the carrier of L opp and
A8: the InternalRel of S c= the InternalRel of L opp;
  thus the carrier of S opp c= the carrier of L & the InternalRel of S opp c=
  the InternalRel of L by A7,A8,Th1;
end;
