
theorem Th25:
  for L being RelStr for S being full SubRelStr of L for R being
  SubRelStr of S holds R is full iff R is full SubRelStr of L
proof
  let L be RelStr, S be full SubRelStr of L, R be SubRelStr of S;
  reconsider R9 = R as SubRelStr of L by YELLOW_6:7;
A1: the carrier of R c= the carrier of S by YELLOW_0:def 13;
  hereby
    assume R is full;
    then the InternalRel of R = (the InternalRel of S)|_2 the carrier of R
      .= ((the InternalRel of L)|_2 the carrier of S)|_2 the carrier of R by
YELLOW_0:def 14
      .= (the InternalRel of L)|_2 the carrier of R9 by A1,WELLORD1:22;
    hence R is full SubRelStr of L by YELLOW_0:def 14;
  end;
  assume
A2: R is full SubRelStr of L;
  ((the InternalRel of L)|_2 the carrier of S)|_2 the carrier of R = (the
  InternalRel of L)|_2 the carrier of R by A1,WELLORD1:22
    .= the InternalRel of R by A2,YELLOW_0:def 14;
  hence the InternalRel of R = (the InternalRel of S)|_2 the carrier of R by
YELLOW_0:def 14;
end;
