
theorem Th59:
  for L be non empty reflexive transitive RelStr for S be non
  empty full SubRelStr of L holds idsMap S is monotone
proof
  let L be non empty reflexive transitive RelStr;
  let S be non empty full SubRelStr of L;
  set f = idsMap S;
  now
    let x, y be Element of InclPoset Ids S;
    reconsider I = x, J = y as Ideal of S by YELLOW_2:41;
    consider K1 be Subset of L such that
A1: I = K1 and
A2: f.x = downarrow K1 by Def11;
    consider K2 be Subset of L such that
A3: J = K2 and
A4: f.y = downarrow K2 by Def11;
    assume x <= y;
    then I c= J by YELLOW_1:3;
    then downarrow K1 c= downarrow K2 by A1,A3,YELLOW_3:6;
    hence f.x <= f.y by A2,A4,YELLOW_1:3;
  end;
  hence thesis by WAYBEL_1:def 2;
end;
