set P = the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ;
set T = TAS-structure(# the carrier of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adjectives of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the InternalRel of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the non-op of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adj-map of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ,(sub the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ) #);
RelStr(# the carrier of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the InternalRel of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure #) = RelStr(# the carrier of TAS-structure(# the carrier of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adjectives of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the InternalRel of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the non-op of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adj-map of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ,(sub the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ) #), the InternalRel of TAS-structure(# the carrier of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adjectives of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the InternalRel of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the non-op of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adj-map of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ,(sub the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ) #) #) ;
then reconsider T = TAS-structure(# the carrier of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adjectives of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the InternalRel of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the non-op of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the adj-map of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ,(sub the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ) #) as 1 -element reflexive non void strict TAS-structure by Def4, STRUCT_0:def 19, WAYBEL_8:12;
take T ; :: thesis: ( T is Mizar-widening-like & T is involutive & T is without_fixpoints & T is consistent & T is adj-structured & T is adjs-typed & T is non-absorbing & T is subjected & T is commutative )
thus T is Mizar-widening-like ; :: thesis: ( T is involutive & T is without_fixpoints & T is consistent & T is adj-structured & T is adjs-typed & T is non-absorbing & T is subjected & T is commutative )
AdjectiveStr(# the adjectives of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the non-op of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure #) = AdjectiveStr(# the adjectives of T, the non-op of T #) ;
hence ( T is involutive & T is without_fixpoints ) by Th5, Th6; :: thesis: ( T is consistent & T is adj-structured & T is adjs-typed & T is non-absorbing & T is subjected & T is commutative )
thus ( T is consistent & T is adj-structured & T is adjs-typed ) by Th8, Th9, Th17; :: thesis: ( T is non-absorbing & T is subjected & T is commutative )

hereby :: according to ABCMIZ_0:def 26 :: thesis: ( T is subjected & T is commutative )

let a be adjective of T; :: thesis: sub (non- a) = sub a
reconsider b = a as adjective of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ;
thus sub (non- a) = sup ((types (non- b)) \/ (types (non- (non- b)))) by Def22
.= sup ((types (non- b)) \/ (types b)) by Def6
.= sub a by Def22 ; :: thesis: verum

end;

A1: RelStr(# the carrier of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure , the InternalRel of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure #) = RelStr(# the carrier of T, the InternalRel of T #) ;
thus T is subjected :: thesis: T is commutative

proof

let a be adjective of T; :: according to ABCMIZ_0:def 25 :: thesis: ( (types a) \/ (types (non- a)) is_<=_than sub a & ( types a <> {} & types (non- a) <> {} implies sub a = sup ((types a) \/ (types (non- a))) ) )
reconsider b = a as adjective of the 1 -element reflexive complete Mizar-widening-like non void involutive without_fixpoints strict consistent adj-structured adjs-typed TA-structure ;
A2: types (non- a) = types (non- b) by Th11;
types a = types b by Th11;
then sup ((types a) \/ (types (non- a))) = sup ((types b) \/ (types (non- b))) by A1, A2, YELLOW_0:17, YELLOW_0:26;
then sup ((types a) \/ (types (non- a))) = sub a by Def22;
hence ( (types a) \/ (types (non- a)) is_<=_than sub a & ( types a <> {} & types (non- a) <> {} implies sub a = sup ((types a) \/ (types (non- a))) ) ) by YELLOW_0:32; :: thesis: verum

end;

let t1, t2 be type of T; :: according to ABCMIZ_0:def 30 :: thesis: for a being adjective of T st a is_properly_applicable_to t1 & a ast t1 <= t2 holds
ex A being finite Subset of the adjectives of T st
( A is_properly_applicable_to t1 "\/" t2 & A ast (t1 "\/" t2) = t2 )
let a be adjective of T; :: thesis: ( a is_properly_applicable_to t1 & a ast t1 <= t2 implies ex A being finite Subset of the adjectives of T st
( A is_properly_applicable_to t1 "\/" t2 & A ast (t1 "\/" t2) = t2 ) )
assume that
a is_properly_applicable_to t1 and
a ast t1 <= t2 ; :: thesis: ex A being finite Subset of the adjectives of T st
( A is_properly_applicable_to t1 "\/" t2 & A ast (t1 "\/" t2) = t2 )
take A = {} the adjectives of T; :: thesis: ( A is_properly_applicable_to t1 "\/" t2 & A ast (t1 "\/" t2) = t2 )
thus A is_properly_applicable_to t1 "\/" t2 by Th63; :: thesis: A ast (t1 "\/" t2) = t2
thus A ast (t1 "\/" t2) = t2 by STRUCT_0:def 10; :: thesis: verum