
theorem Th9:
  for S being RelStr, T being non empty reflexive RelStr, x being
  set holds x is Element of MonMaps(S,T) iff x is monotone Function of S,T
proof
  let S be RelStr, T be non empty reflexive RelStr;
  let x be set;
  hereby
    assume x is Element of MonMaps(S,T);
    then reconsider f = x as Element of MonMaps(S,T);
    f is Element of T|^the carrier of S by YELLOW_0:58;
    then f in the carrier of T|^the carrier of S;
    then f in Funcs(the carrier of S, the carrier of T) by YELLOW_1:28;
    then reconsider f as Function of S,T by FUNCT_2:66;
    f in the carrier of MonMaps(S,T);
    hence x is monotone Function of S,T by YELLOW_1:def 6;
  end;
  assume x is monotone Function of S,T;
  then reconsider f = x as monotone Function of S,T;
  f in Funcs(the carrier of S, the carrier of T) by FUNCT_2:8;
  hence thesis by YELLOW_1:def 6;
end;
