
theorem Th12:
  for L be non empty RelStr for S be non empty SubRelStr of L for
  X be Subset of L for Y be Subset of S st X = Y holds uparrow Y c= uparrow X
proof
  let L be non empty RelStr;
  let S be non empty SubRelStr of L;
  let X be Subset of L;
  let Y be Subset of S;
  assume
A1: X = Y;
  let x be object;
  assume
A2: x in uparrow Y;
  then reconsider x1 = x as Element of S;
  consider y1 be Element of S such that
A3: y1 <= x1 and
A4: y1 in Y by A2,WAYBEL_0:def 16;
  reconsider x2 = x1, y2 = y1 as Element of L by YELLOW_0:58;
  y2 <= x2 by A3,YELLOW_0:59;
  hence thesis by A1,A4,WAYBEL_0:def 16;
end;
