
theorem Th32:
  for L being complete LATTICE
  for S being sups-inheriting non empty full SubRelStr of L
  for x,y being Element of L, a,b being Element of S st a = x & b = y
  holds x << y implies a << b
proof
  let L be complete LATTICE;
  let S be sups-inheriting non empty full SubRelStr of L;
  let x,y be Element of L, a,b be Element of S such that
A1: a = x and
A2: b = y and
A3: for D being non empty directed Subset of L st y <= sup D
  ex d being Element of L st d in D & x <= d;
  let D be non empty directed Subset of S such that
A4: b <= sup D;
  reconsider E = D as non empty directed Subset of L by YELLOW_2:7;
A5: ex_sup_of D, L by YELLOW_0:17;
  then "\/"(D,L) in the carrier of S by YELLOW_0:def 19;
  then sup E = sup D by A5,YELLOW_0:64;
  then y <= sup E by A2,A4,YELLOW_0:59;
  then consider e being Element of L such that
A6: e in E and
A7: x <= e by A3;
  reconsider d = e as Element of S by A6;
  take d;
  thus thesis by A1,A6,A7,YELLOW_0:60;
end;
