theorem
  s ==>. t, S implies S, (S \/ {[s, t]}) are_equivalent_wrt w
proof
  assume s ==>. t, S;
  then [s, t] in ==>.-relation(S) by Def6;
  then {[s, t]} c= ==>.-relation(S) by ZFMISC_1:31;
  then
A1: S \/ {[s, t]} c= S \/ ==>.-relation(S) by XBOOLE_1:9;
  S, S \/ ==>.-relation(S) are_equivalent_wrt w & S c= S \/ {[s, t]} by Th57
,Th60,XBOOLE_1:7;
  hence thesis by A1,Th56;
end;
