
theorem Th27:
  for R being non empty RelStr, X being non empty Subset of R
  holds the mapping of X+id = id X & the mapping of X opp+id = id X
proof
  let R be non empty RelStr, X be non empty Subset of R;
A1: now
    let x be object;
    assume x in X;
    then reconsider i = x as Element of X+id by YELLOW_9:6;
    thus (the mapping of X+id).x = X+id.i .= x by YELLOW_9:6;
  end;
  the carrier of X+id = X by YELLOW_9:6;
  then dom the mapping of X+id = X by FUNCT_2:def 1;
  hence the mapping of X+id = id X by A1,FUNCT_1:17;
A2: now
    let x be object;
    assume x in X;
    then reconsider i = x as Element of X opp+id by YELLOW_9:7;
    thus (the mapping of X opp+id).x = X opp+id.i .= x by YELLOW_9:7;
  end;
  the carrier of X opp+id = X by YELLOW_9:7;
  then dom the mapping of X opp+id = X by FUNCT_2:def 1;
  hence thesis by A2,FUNCT_1:17;
end;
