
theorem Th12:
  for T1,T2,R being RelStr, S being SubRelStr of T1
  st the RelStr of T1 = the RelStr of T2 & the RelStr of S = the RelStr of R
  holds R is SubRelStr of T2 & (S is full implies R is full SubRelStr of T2)
proof
  let T,T2,R be RelStr, S be SubRelStr of T such that
A1: the RelStr of T = the RelStr of T2 and
A2: the RelStr of S = the RelStr of R;
A3: the carrier of R c= the carrier of T2 by A1,A2,YELLOW_0:def 13;
A4: the InternalRel of R c= the InternalRel of T2 by A1,A2,YELLOW_0:def 13;
  hence R is SubRelStr of T2 by A3,YELLOW_0:def 13;
  assume the InternalRel of S = (the InternalRel of T)|_2 the carrier of S;
  hence thesis by A1,A2,A3,A4,YELLOW_0:def 13,def 14;
end;
