
theorem Th3:
  for L,S being RelStr holds (S is full SubRelStr of L iff S opp is
  full SubRelStr of L opp) & (S opp is full SubRelStr of L iff S is full
  SubRelStr of L opp)
proof
  let L,S be RelStr;
A1: ((the InternalRel of L)|_2 the carrier of S)~ = (the InternalRel of L)~
  |_2 the carrier of S by ORDERS_1:83;
  hereby
    assume
A2: S is full SubRelStr of L;
    then the InternalRel of S = (the InternalRel of L)|_2 the carrier of S by
YELLOW_0:def 14;
    hence S opp is full SubRelStr of L opp by A1,A2,Th2,YELLOW_0:def 14;
  end;
A3: ((the InternalRel of L)~|_2 the carrier of S)~ = (the InternalRel of L)~
  ~|_2 the carrier of S by ORDERS_1:83;
  hereby
    assume
A4: S opp is full SubRelStr of L opp;
    then the InternalRel of S opp = (the InternalRel of L opp)|_2 the carrier
    of S by YELLOW_0:def 14;
    hence S is full SubRelStr of L by A3,A4,Th2,YELLOW_0:def 14;
  end;
  hereby
    assume
A5: S opp is full SubRelStr of L;
    then
    the InternalRel of S opp = (the InternalRel of L)|_2 the carrier of S
    by YELLOW_0:def 14;
    hence S is full SubRelStr of L opp by A1,A5,Th2,YELLOW_0:def 14;
  end;
  assume
A6: S is full SubRelStr of L opp;
  then the InternalRel of S = (the InternalRel of L opp)|_2 the carrier of S
  by YELLOW_0:def 14;
  hence thesis by A1,A6,Th2,YELLOW_0:def 14;
end;
