
theorem Th17:
  for L being reflexive transitive RelStr, R being auxiliary(ii) (
  Relation of L), C being Subset of L, x, y being Element of L st x <= y holds
  SetBelow (R,C,x) c= SetBelow (R,C,y)
proof
  let L be reflexive transitive RelStr, R be auxiliary(ii) (Relation of L), C
  be Subset of L, x, y be Element of L such that
A1: x <= y;
  let a be object;
  assume
A2: a in SetBelow (R,C,x);
  then reconsider L as non empty reflexive RelStr;
  reconsider a as Element of L by A2;
A3: a in C by A2,Th15;
A4: a <= a;
  [a,x] in R by A2,Th15;
  then [a,y] in R by A4,A1,WAYBEL_4:def 4;
  hence thesis by A3,Th15;
end;
