
theorem Th61:
  for L being non empty RelStr, S being non empty full SubRelStr
  of L for X being set, a being Element of L for x being Element of S st x = a
  holds (a is_<=_than X implies x is_<=_than X) & (a is_>=_than X implies x
  is_>=_than X)
proof
  let L be non empty RelStr, S be non empty full SubRelStr of L,X be set;
  let a be Element of L;
  let x be Element of S;
  assume
A1: x = a;
  hereby
    assume
A2: a is_<=_than X;
    thus x is_<=_than X
    proof
      let y be Element of S;
      reconsider b = y as Element of L by Th58;
      assume y in X;
      then a <= b by A2;
      hence thesis by A1,Th60;
    end;
  end;
  assume
A3: a is_>=_than X;
  let y be Element of S;
  reconsider b = y as Element of L by Th58;
  assume y in X;
  then a >= b by A3;
  hence thesis by A1,Th60;
end;
