
theorem Th60:
  for L being RelStr, S being full SubRelStr of L for a,b being
  Element of L for x,y being Element of S st x = a & y = b & a <= b & x in the
  carrier of S holds x <= y
proof
  let L be RelStr, S be full SubRelStr of L;
  let a,b be Element of L, x,y be Element of S such that
A1: x = a & y = b;
  assume
A2: [a,b] in the InternalRel of L;
A3: the InternalRel of S = (the InternalRel of L)|_2 the carrier of S by Def14;
  assume x in the carrier of S;
  then [x,y] in [:the carrier of S,the carrier of S:] by ZFMISC_1:87;
  hence [x,y] in the InternalRel of S by A1,A3,A2,XBOOLE_0:def 4;
end;
