
theorem Th77:
  for T being adj-structured antisymmetric commutative non void
  reflexive transitive with_suprema Noetherian TAS-structure holds T@--> is
  with_Church-Rosser_property with_UN_property
proof
  let T be adj-structured with_suprema antisymmetric commutative non empty
  non void reflexive transitive Noetherian TAS-structure;
  set R = T@-->;
  R is locally-confluent
  proof
    let a,b,c be object;
    assume that
A1: [a,b] in R and
A2: [a,c] in R;
    reconsider t = a, t1 = b, t2 = c as type of T by A1,A2,ZFMISC_1:87;
    consider a2 being adjective of T such that
    not a2 in adjs t1 and
A3: a2 is_properly_applicable_to t1 and
A4: a2 ast t1 = t by A1,Def31;
    set tt = t1 "\/" t2;
    take tt;
    consider a3 being adjective of T such that
    not a3 in adjs t2 and
A5: a3 is_properly_applicable_to t2 and
A6: a3 ast t2 = t by A2,Def31;
    a3 is_applicable_to t2 by A5;
    then t <= t2 by A6,Th20;
    then
A7: ex B being finite Subset of the adjectives of T st B
    is_properly_applicable_to t1 "\/" t2 & B ast (t1 "\/" t2) = t2 by A3,A4
,Def30;
    a2 is_applicable_to t1 by A3;
    then t <= t1 by A4,Th20;
    then ex A being finite Subset of the adjectives of T st A
    is_properly_applicable_to t1 "\/" t2 & A ast (t1 "\/" t2) = t1 by A5,A6
,Def30;
    hence thesis by A7,Th72;
  end;
  then R is strongly-normalizing locally-confluent Relation by Th69;
  hence thesis;
end;
