d1_abcmiz_0:: for T being RelStr holds ( T is Noetherian iff the InternalRel of T is co-well_founded );
d10_abcmiz_0:: for T being non empty TA-structure holds ( T is adj-structured iff the adj-map of T is join-preserving Function of T,((BoolePoset the adjectives of T) opp) );
d11_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema TA-structure holds ( T is adj-structured iff for t1, t2 being type of T holds adjs (t1 "\/" t2) = (adjs t1) /\ (adjs t2) );
d12_abcmiz_0:: for T being TA-structure for a being Element of the adjectives of T for b3 being Subset of T holds ( b3 = types a iff for x being set holds ( x in b3 iff ex t being type of T st ( x = t & a in adjs t ) ) );
d13_abcmiz_0:: for T being non empty TA-structure for A being Subset of the adjectives of T for b3 being Subset of T holds ( b3 = types A iff for t being type of T holds ( t in b3 iff for a being adjective of T st a in A holds t in types a ) );
d14_abcmiz_0:: for T being TA-structure holds ( T is adjs-typed iff for a being adjective of T holds not (types a) \/ (types (non- a)) is empty );
d15_abcmiz_0:: for T being TA-structure for t being Element of T for a being adjective of T holds ( a is_applicable_to t iff ex t9 being type of T st ( t9 in types a & t9 <= t ) );
d16_abcmiz_0:: for T being TA-structure for t being type of T for A being Subset of the adjectives of T holds ( A is_applicable_to t iff ex t9 being type of T st ( A c= adjs t9 & t9 <= t ) );
d17_abcmiz_0:: for T being non empty reflexive transitive TA-structure for t being Element of T for a being adjective of T holds a ast t = sup ((types a) /\ (downarrow t));
d18_abcmiz_0:: for T being non empty reflexive transitive TA-structure for t being type of T for A being Subset of the adjectives of T holds A ast t = sup ((types A) /\ (downarrow t));
d19_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for p being FinSequence of the adjectives of T for b4 being FinSequence of the carrier of T holds ( b4 = apply (p,t) iff ( len b4 = (len p) + 1 & b4 . 1 = t & ( for i being Element of NAT for a being adjective of T for t being type of T st i in dom p & a = p . i & t = b4 . i holds b4 . (i + 1) = a ast t ) ) );
d2_abcmiz_0:: for T being non empty RelStr holds ( T is Noetherian iff for A being non empty Subset of T ex a being Element of T st ( a in A & ( for b being Element of T st b in A holds not a < b ) ) );
d20_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for v being FinSequence of the adjectives of T holds v ast t = (apply (v,t)) . ((len v) + 1);
d21_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for v being FinSequence of the adjectives of T holds ( v is_applicable_to t iff for i being Nat for a being adjective of T for s being type of T st i in dom v & a = v . i & s = (apply (v,t)) . i holds a is_applicable_to s );
d22_abcmiz_0:: for T being non empty non void TA-structure for b2 being Function of the adjectives of T, the carrier of T holds ( b2 = sub T iff for a being adjective of T holds b2 . a = sup ((types a) \/ (types (non- a))) );
d23_abcmiz_0:: for T being non empty non void TAS-structure for a being adjective of T holds sub a = the sub-map of T . a;
d24_abcmiz_0:: for T being non empty non void TAS-structure holds ( T is non-absorbing iff the sub-map of T * the non-op of T = the sub-map of T );
d25_abcmiz_0:: for T being non empty non void TAS-structure holds ( T is subjected iff for a being adjective of T holds ( (types a) \/ (types (non- a)) is_<=_than sub a & ( types a <> {} & types (non- a) <> {} implies sub a = sup ((types a) \/ (types (non- a))) ) ) );
d26_abcmiz_0:: for T being non empty non void TAS-structure holds ( T is non-absorbing iff for a being adjective of T holds sub (non- a) = sub a );
d27_abcmiz_0:: for T being non empty non void TAS-structure for t being Element of T for a being adjective of T holds ( a is_properly_applicable_to t iff ( t <= sub a & a is_applicable_to t ) );
d28_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for v being FinSequence of the adjectives of T holds ( v is_properly_applicable_to t iff for i being Nat for a being adjective of T for s being type of T st i in dom v & a = v . i & s = (apply (v,t)) . i holds a is_properly_applicable_to s );
d29_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for A being Subset of the adjectives of T holds ( A is_properly_applicable_to t iff ex s being FinSequence of the adjectives of T st ( rng s = A & s is_properly_applicable_to t ) );
d3_abcmiz_0:: for T being Poset holds ( T is Mizar-widening-like iff ( T is sup-Semilattice & T is Noetherian ) );
d30_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure holds ( T is commutative iff for t1, t2 being type of T 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 ) );
d31_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for b2 being Relation of T holds ( b2 = T @--> iff for t1, t2 being type of T holds ( [t1,t2] in b2 iff ex a being adjective of T st ( not a in adjs t2 & a is_properly_applicable_to t2 & a ast t2 = t1 ) ) );
d32_abcmiz_0:: for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure for t being type of T holds radix t = nf (t,(T @-->));
d4_abcmiz_0:: for A being AdjectiveStr holds ( A is void iff the adjectives of A is empty );
d5_abcmiz_0:: for A being AdjectiveStr for a being Element of the adjectives of A holds non- a = the non-op of A . a;
d6_abcmiz_0:: for A being AdjectiveStr holds ( A is involutive iff for a being adjective of A holds non- (non- a) = a );
d7_abcmiz_0:: for A being AdjectiveStr holds ( A is without_fixpoints iff for a being adjective of A holds not non- a = a );
d8_abcmiz_0:: for T being TA-structure for t being Element of T holds adjs t = the adj-map of T . t;
d9_abcmiz_0:: for T being TA-structure holds ( T is consistent iff for t being type of T for a being adjective of T st a in adjs t holds not non- a in adjs t );
s1_abcmiz_0:: scheme MinimalFiniteSet{ P1[ set ] } : ex A being finite set st ( P1[A] & ( for B being set st B c= A & P1[B] holds B = A ) ) provided A1: ex A being finite set st P1[A]
s2_abcmiz_0:: scheme RedInd{ F1() -> non empty set , P1[ set , set ], F2() -> Relation of F1() } : for x, y being Element of F1() st F2() reduces x,y holds P1[x,y] provided A1: for x, y being Element of F1() st [x,y] in F2() holds P1[x,y] and A2: for x being Element of F1() holds P1[x,x] and A3: for x, y, z being Element of F1() st P1[x,y] & P1[y,z] holds P1[x,z]
t1_abcmiz_0:: for T being Noetherian sup-Semilattice for I being Ideal of T holds ( ex_sup_of I,T & sup I in I )
t10_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema TA-structure st T is adj-structured holds for t1, t2 being type of T st t1 <= t2 holds adjs t2 c= adjs t1
t11_abcmiz_0:: for T1, T2 being TA-structure st TA-structure(# the carrier of T1, the adjectives of T1, the InternalRel of T1, the non-op of T1, the adj-map of T1 #) = TA-structure(# the carrier of T2, the adjectives of T2, the InternalRel of T2, the non-op of T2, the adj-map of T2 #) holds for a1 being adjective of T1 for a2 being adjective of T2 st a1 = a2 holds types a1 = types a2
t12_abcmiz_0:: for T being non empty TA-structure for a being adjective of T holds types a = { t where t is type of T : a in adjs t }
t13_abcmiz_0:: for T being TA-structure for t being type of T for a being adjective of T holds ( a in adjs t iff t in types a )
t14_abcmiz_0:: for T being non empty TA-structure for t being type of T for A being Subset of the adjectives of T holds ( A c= adjs t iff t in types A )
t15_abcmiz_0:: for T being non void TA-structure for t being type of T holds adjs t = { a where a is adjective of T : t in types a }
t16_abcmiz_0:: for T being non empty TA-structure holds types ({} the adjectives of T) = the carrier of T
t17_abcmiz_0:: for T1, T2 being TA-structure st TA-structure(# the carrier of T1, the adjectives of T1, the InternalRel of T1, the non-op of T1, the adj-map of T1 #) = TA-structure(# the carrier of T2, the adjectives of T2, the InternalRel of T2, the non-op of T2, the adj-map of T2 #) & T1 is adjs-typed holds T2 is adjs-typed
t18_abcmiz_0:: for T being consistent TA-structure for a being adjective of T holds types a misses types (non- a)
t19_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema adj-structured TA-structure for a being adjective of T for t being type of T st a is_applicable_to t holds (types a) /\ (downarrow t) is Ideal of T
t2_abcmiz_0:: for A1, A2 being AdjectiveStr st the adjectives of A1 = the adjectives of A2 & A1 is void holds A2 is void
t20_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for a being adjective of T st a is_applicable_to t holds a ast t <= t
t21_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for a being adjective of T st a is_applicable_to t holds a in adjs (a ast t)
t22_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for a being adjective of T st a is_applicable_to t holds a ast t in types a
t23_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for a being adjective of T for t9 being type of T st t9 <= t & a in adjs t9 holds ( a is_applicable_to t & t9 <= a ast t )
t24_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for a being adjective of T st a in adjs t holds ( a is_applicable_to t & a ast t = t )
t25_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for a, b being adjective of T st a is_applicable_to t & b is_applicable_to a ast t holds ( b is_applicable_to t & a is_applicable_to b ast t & a ast (b ast t) = b ast (a ast t) )
t26_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema adj-structured TA-structure for A being Subset of the adjectives of T for t being type of T st A is_applicable_to t holds (types A) /\ (downarrow t) is Ideal of T
t27_abcmiz_0:: for T being non empty reflexive transitive antisymmetric TA-structure for t being type of T holds ({} the adjectives of T) ast t = t
t28_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T holds apply ((<*> the adjectives of T),t) = <*t*>
t29_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for a being adjective of T holds apply (<*a*>,t) = <*t,(a ast t)*>
t3_abcmiz_0:: for A1, A2 being AdjectiveStr st AdjectiveStr(# the adjectives of A1, the non-op of A1 #) = AdjectiveStr(# the adjectives of A2, the non-op of A2 #) holds for a1 being adjective of A1 for a2 being adjective of A2 st a1 = a2 holds non- a1 = non- a2 ;
t30_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T holds (<*> the adjectives of T) ast t = t by Def19;
t31_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for a being adjective of T holds <*a*> ast t = a ast t
t32_abcmiz_0:: for p, q being FinSequence for i being Nat st i >= 1 & i < len p holds (p $^ q) . i = p . i
t33_abcmiz_0:: for p being non empty FinSequence for q being FinSequence for i being Nat st i < len q holds (p $^ q) . ((len p) + i) = q . (i + 1)
t34_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T holds apply ((v1 ^ v2),t) = (apply (v1,t)) $^ (apply (v2,(v1 ast t)))
t35_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T for i being Nat st i in dom v1 holds (apply ((v1 ^ v2),t)) . i = (apply (v1,t)) . i
t36_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T holds (apply ((v1 ^ v2),t)) . ((len v1) + 1) = v1 ast t
t37_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T holds v2 ast (v1 ast t) = (v1 ^ v2) ast t
t38_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T holds <*> the adjectives of T is_applicable_to t
t39_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for a being adjective of T holds ( a is_applicable_to t iff <*a*> is_applicable_to t )
t4_abcmiz_0:: for a1, a2 being set st a1 <> a2 holds for A being AdjectiveStr st the adjectives of A = {a1,a2} & the non-op of A . a1 = a2 & the non-op of A . a2 = a1 holds ( not A is void & A is involutive & A is without_fixpoints )
t40_abcmiz_0:: for T being non empty reflexive transitive non void TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_applicable_to t holds ( v1 is_applicable_to t & v2 is_applicable_to v1 ast t )
t41_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds for i1, i2 being Nat st 1 <= i1 & i1 <= i2 & i2 <= (len v) + 1 holds for t1, t2 being type of T st t1 = (apply (v,t)) . i1 & t2 = (apply (v,t)) . i2 holds t2 <= t1
t42_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds for s being type of T st s in rng (apply (v,t)) holds ( v ast t <= s & s <= t )
t43_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds v ast t <= t
t44_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds rng v c= adjs (v ast t)
t45_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds for A being Subset of the adjectives of T st A = rng v holds A is_applicable_to t
t46_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1 is_applicable_to t & rng v2 c= rng v1 holds for s being type of T st s in rng (apply (v2,t)) holds v1 ast t <= s
t47_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_applicable_to t holds v2 ^ v1 is_applicable_to t
t48_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_applicable_to t holds (v1 ^ v2) ast t = (v2 ^ v1) ast t
t49_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for A being Subset of the adjectives of T st A is_applicable_to t holds A ast t <= t
t5_abcmiz_0:: for A1, A2 being AdjectiveStr st AdjectiveStr(# the adjectives of A1, the non-op of A1 #) = AdjectiveStr(# the adjectives of A2, the non-op of A2 #) & A1 is involutive holds A2 is involutive
t50_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for A being Subset of the adjectives of T st A is_applicable_to t holds A c= adjs (A ast t)
t51_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for A being Subset of the adjectives of T st A is_applicable_to t holds A ast t in types A
t52_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for A being Subset of the adjectives of T for t9 being type of T st t9 <= t & A c= adjs t9 holds ( A is_applicable_to t & t9 <= A ast t )
t53_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian adj-structured TA-structure for t being type of T for A being Subset of the adjectives of T st A c= adjs t holds ( A is_applicable_to t & A ast t = t )
t54_abcmiz_0:: for T being TA-structure for t being type of T for A, B being Subset of the adjectives of T st A is_applicable_to t & B c= A holds B is_applicable_to t
t55_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for a being adjective of T for A, B being Subset of the adjectives of T st B = A \/ {a} & B is_applicable_to t holds a ast (A ast t) = B ast t
t56_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TA-structure for t being type of T for v being FinSequence of the adjectives of T st v is_applicable_to t holds for A being Subset of the adjectives of T st A = rng v holds v ast t = A ast t
t57_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for v being FinSequence of the adjectives of T st v is_properly_applicable_to t holds v is_applicable_to t
t58_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T holds <*> the adjectives of T is_properly_applicable_to t
t59_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for a being adjective of T holds ( a is_properly_applicable_to t iff <*a*> is_properly_applicable_to t )
t6_abcmiz_0:: for A1, A2 being AdjectiveStr st AdjectiveStr(# the adjectives of A1, the non-op of A1 #) = AdjectiveStr(# the adjectives of A2, the non-op of A2 #) & A1 is without_fixpoints holds A2 is without_fixpoints
t60_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1 ^ v2 is_properly_applicable_to t holds ( v1 is_properly_applicable_to t & v2 is_properly_applicable_to v1 ast t )
t61_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for v1, v2 being FinSequence of the adjectives of T st v1 is_properly_applicable_to t & v2 is_properly_applicable_to v1 ast t holds v1 ^ v2 is_properly_applicable_to t
t62_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for A being Subset of the adjectives of T st A is_properly_applicable_to t holds A is finite
t63_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T holds {} the adjectives of T is_properly_applicable_to t
t64_abcmiz_0:: for T being non empty reflexive transitive non void TAS-structure for t being type of T for A being Subset of the adjectives of T st A is_properly_applicable_to t holds ex B being Subset of the adjectives of T st ( B c= A & B is_properly_applicable_to t & A ast t = B ast t & ( for C being Subset of the adjectives of T st C c= B & C is_properly_applicable_to t & A ast t = C ast t holds C = B ) )
t65_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure for t being type of T for A being Subset of the adjectives of T st A is_properly_applicable_to t holds ex s being one-to-one FinSequence of the adjectives of T st ( rng s = A & s is_properly_applicable_to t )
t66_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds T @--> c= the InternalRel of T
t67_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure for t1, t2 being type of T st T @--> reduces t1,t2 holds t1 <= t2
t68_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds T @--> is irreflexive
t69_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure holds T @--> is strongly-normalizing
t7_abcmiz_0:: for T1, T2 being TA-structure st TA-structure(# the carrier of T1, the adjectives of T1, the InternalRel of T1, the non-op of T1, the adj-map of T1 #) = TA-structure(# the carrier of T2, the adjectives of T2, the InternalRel of T2, the non-op of T2, the adj-map of T2 #) holds for t1 being type of T1 for t2 being type of T2 st t1 = t2 holds adjs t1 = adjs t2 ;
t70_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure for t being type of T for A being finite Subset of the adjectives of T st ( for C being Subset of the adjectives of T st C c= A & C is_properly_applicable_to t & A ast t = C ast t holds C = A ) holds for s being one-to-one FinSequence of the adjectives of T st rng s = A & s is_properly_applicable_to t holds for i being Nat st 1 <= i & i <= len s holds [((apply (s,t)) . (i + 1)),((apply (s,t)) . i)] in T @-->
t71_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure for t being type of T for A being finite Subset of the adjectives of T st ( for C being Subset of the adjectives of T st C c= A & C is_properly_applicable_to t & A ast t = C ast t holds C = A ) holds for s being one-to-one FinSequence of the adjectives of T st rng s = A & s is_properly_applicable_to t holds Rev (apply (s,t)) is RedSequence of T @-->
t72_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure for t being type of T for A being finite Subset of the adjectives of T st A is_properly_applicable_to t holds T @--> reduces A ast t,t
t73_abcmiz_0:: for X being non empty set for R being Relation of X for r being RedSequence of R st r . 1 in X holds r is FinSequence of X
t74_abcmiz_0:: for X being non empty set for R being Relation of X for x being Element of X for y being set st R reduces x,y holds y in X
t75_abcmiz_0:: for X being non empty set for R being Relation of X st R is with_UN_property & R is weakly-normalizing holds for x being Element of X holds nf (x,R) in X
t76_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured TAS-structure for t1, t2 being type of T st T @--> reduces t1,t2 holds ex A being finite Subset of the adjectives of T st ( A is_properly_applicable_to t2 & t1 = A ast t2 )
t77_abcmiz_0:: for T being reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure holds ( T @--> is with_Church-Rosser_property & T @--> is with_UN_property )
t78_abcmiz_0:: for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure for t being type of T holds T @--> reduces t, radix t
t79_abcmiz_0:: for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure for t being type of T holds t <= radix t by Th67, Th78;
t8_abcmiz_0:: for T1, T2 being TA-structure st TA-structure(# the carrier of T1, the adjectives of T1, the InternalRel of T1, the non-op of T1, the adj-map of T1 #) = TA-structure(# the carrier of T2, the adjectives of T2, the InternalRel of T2, the non-op of T2, the adj-map of T2 #) & T1 is consistent holds T2 is consistent
t80_abcmiz_0:: for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure for t being type of T for X being set st X = { t9 where t9 is type of T : ex A being finite Subset of the adjectives of T st ( A is_properly_applicable_to t9 & A ast t9 = t ) } holds ( ex_sup_of X,T & radix t = "\/" (X,T) )
t81_abcmiz_0:: for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure for t1, t2 being type of T for a being adjective of T st a is_properly_applicable_to t1 & a ast t1 <= radix t2 holds t1 <= radix t2
t82_abcmiz_0:: for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure for t1, t2 being type of T st t1 <= t2 holds radix t1 <= radix t2
t83_abcmiz_0:: for T being non empty reflexive transitive antisymmetric with_suprema Noetherian non void adj-structured commutative TAS-structure for t being type of T for a being adjective of T st a is_properly_applicable_to t holds radix (a ast t) = radix t
t9_abcmiz_0:: for T1, T2 being non empty TA-structure st TA-structure(# the carrier of T1, the adjectives of T1, the InternalRel of T1, the non-op of T1, the adj-map of T1 #) = TA-structure(# the carrier of T2, the adjectives of T2, the InternalRel of T2, the non-op of T2, the adj-map of T2 #) & T1 is adj-structured holds T2 is adj-structured
d1_abcmiz_1:: for A being set for b2 being set holds ( b2 = varcl A iff ( A c= b2 & ( for x, y being set st [x,y] in b2 holds x c= b2 ) & ( for B being set st A c= B & ( for x, y being set st [x,y] in B holds x c= B ) holds b2 c= B ) ) );
d10_abcmiz_1:: for b1 being strict Signature holds ( b1 = MinConstrSign iff ( b1 is constructor & the carrier' of b1 = {*,non_op} ) );
d11_abcmiz_1:: for C being ConstructorSignature for o being OperSymbol of C holds ( o is constructor iff ( o <> * & o <> non_op ) );
d12_abcmiz_1:: for C being non empty non void Signature holds ( C is initialized iff ex m, a being OperSymbol of C st ( the_result_sort_of m = a_Type & the_arity_of m = {} & the_result_sort_of a = an_Adj & the_arity_of a = {} ) );
d13_abcmiz_1:: for C being ConstructorSignature holds a_Type C = a_Type ;
d14_abcmiz_1:: for C being ConstructorSignature holds an_Adj C = an_Adj ;
d15_abcmiz_1:: for C being ConstructorSignature holds a_Term C = a_Term ;
d16_abcmiz_1:: for C being ConstructorSignature holds non_op C = non_op ;
d17_abcmiz_1:: for C being ConstructorSignature holds ast C = * ;
d18_abcmiz_1:: Modes = [:{a_Type},[:QuasiLoci,NAT:]:];
d19_abcmiz_1:: Attrs = [:{an_Adj},[:QuasiLoci,NAT:]:];
d2_abcmiz_1:: for b1 being set holds ( b1 = Vars iff ex V being ManySortedSet of NAT st ( b1 = Union V & V . 0 = { [{},i] where i is Element of NAT : verum } & ( for n being Nat holds V . (n + 1) = { [(varcl A),j] where A is Subset of (V . n), j is Element of NAT : A is finite } ) ) );
d20_abcmiz_1:: Funcs = [:{a_Term},[:QuasiLoci,NAT:]:];
d21_abcmiz_1:: Constructors = (Modes \/ Attrs) \/ Funcs;
d22_abcmiz_1:: for c being Element of Constructors holds loci_of c = (c `2) `1 ;
d23_abcmiz_1:: for c being Element of Constructors holds index_of c = (c `2) `2 ;
d24_abcmiz_1:: for b1 being strict ConstructorSignature holds ( b1 = MaxConstrSign iff ( the carrier' of b1 = {*,non_op} \/ Constructors & ( for o being OperSymbol of b1 st o is constructor holds ( the ResultSort of b1 . o = o `1 & card ( the Arity of b1 . o) = card ((o `2) `1) ) ) ) );
d25_abcmiz_1:: for C being ConstructorSignature for b2 being ManySortedSet of the carrier of C holds ( b2 = MSVars C iff ( b2 . a_Type = {} & b2 . an_Adj = {} & b2 . a_Term = Vars ) );
d26_abcmiz_1:: for S being non void Signature for X being V9() ManySortedSet of the carrier of S for t being Element of (Free (S,X)) holds ( t is ground iff Union (S variables_in t) = {} );
d27_abcmiz_1:: for S being non void Signature for X being V9() ManySortedSet of the carrier of S for t being Element of (Free (S,X)) holds ( t is compound iff t . {} in [: the carrier' of S,{ the carrier of S}:] );
d28_abcmiz_1:: for C being initialized ConstructorSignature for s being SortSymbol of C for b3 being expression of C holds ( b3 is expression of C,s iff b3 in the Sorts of (Free (C,(MSVars C))) . s );
d29_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C st len (the_arity_of c) = 0 holds c term = [c, the carrier of C] -tree {};
d3_abcmiz_1:: for b1 being FinSequenceSet of Vars holds ( b1 = QuasiLoci iff for p being FinSequence of Vars holds ( p in b1 iff ( p is one-to-one & ( for i being Nat st i in dom p holds (p . i) `1 c= rng (p dom i) ) ) ) );
d30_abcmiz_1:: for C being initialized ConstructorSignature for o being OperSymbol of C st len (the_arity_of o) = 1 holds for e being expression of C st ex s being SortSymbol of C st ( s = (the_arity_of o) . 1 & e is expression of C,s ) holds o term e = [o, the carrier of C] -tree <*e*>;
d31_abcmiz_1:: for C being initialized ConstructorSignature for o being OperSymbol of C st len (the_arity_of o) = 2 holds for e1, e2 being expression of C st ex s1, s2 being SortSymbol of C st ( s1 = (the_arity_of o) . 1 & s2 = (the_arity_of o) . 2 & e1 is expression of C,s1 & e2 is expression of C,s2 ) holds o term (e1,e2) = [o, the carrier of C] -tree <*e1,e2*>;
d32_abcmiz_1:: for S being non void Signature for s being SortSymbol of S st ex o being OperSymbol of S st the_result_sort_of o = s holds for b3 being OperSymbol of S holds ( b3 is OperSymbol of s iff the_result_sort_of b3 = s );
d33_abcmiz_1:: for C being initialized ConstructorSignature holds QuasiTerms C = the Sorts of (Free (C,(MSVars C))) . (a_Term C);
d34_abcmiz_1:: for x being variable for C being initialized ConstructorSignature holds x -term C = root-tree [x,a_Term];
d35_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len (the_arity_of c) holds c -trm p = [c, the carrier of C] -tree p;
d36_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds ( ( ex a9 being expression of C, an_Adj C st a = (non_op C) term a9 implies Non a = a | <*0*> ) & ( ( for a9 being expression of C, an_Adj C holds not a = (non_op C) term a9 ) implies Non a = (non_op C) term a ) );
d37_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds ( a is positive iff for a9 being expression of C, an_Adj C holds not a = (non_op C) term a9 );
d38_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds ( a is negative iff ex a9 being expression of C, an_Adj C st ( a9 is positive & a = (non_op C) term a9 ) );
d39_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds ( a is regular iff ( a is positive or a is negative ) );
d4_abcmiz_1:: a_Type = 0 ;
d40_abcmiz_1:: for C being initialized ConstructorSignature holds QuasiAdjs C = { a where a is expression of C, an_Adj C : a is regular } ;
d41_abcmiz_1:: for C being initialized ConstructorSignature for q being expression of C, a_Type C holds ( q is pure iff for a being expression of C, an_Adj C for t being expression of C, a_Type C holds not q = (ast C) term (a,t) );
d42_abcmiz_1:: for C being initialized ConstructorSignature holds QuasiTypes C = { [A,t] where t is expression of C, a_Type C, A is finite Subset of (QuasiAdjs C) : t is pure } ;
d43_abcmiz_1:: for C being initialized ConstructorSignature for b2 being set holds ( b2 is quasi-type of C iff b2 in QuasiTypes C );
d44_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C for a being quasi-adjective of C holds a ast T = [({a} \/ (adjs T)),(the_base_of T)];
d45_abcmiz_1:: for S being non void Signature for X being V9() ManySortedSet of the carrier of S for s being SortSymbol of S for b4 being Function of (Union the Sorts of (Free (S,X))),(bool (X . s)) holds ( b4 = (X,s) variables_in iff for t being Element of (Free (S,X)) holds b4 . t = (S variables_in t) . s );
d46_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C holds variables_in e = (C variables_in e) . (a_Term C);
d47_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C holds vars e = varcl (variables_in e);
d48_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C holds variables_in T = (union ((((MSVars C),(a_Term C)) variables_in) .: (adjs T))) \/ (variables_in (the_base_of T));
d49_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C holds vars T = varcl (variables_in T);
d5_abcmiz_1:: an_Adj = 1;
d50_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C holds ( T is ground iff variables_in T = {} );
d51_abcmiz_1:: VarPoset = (InclPoset { (varcl A) where A is finite Subset of Vars : verum } ) opp ;
d52_abcmiz_1:: for C being initialized ConstructorSignature for b2 being Function of (QuasiTypes C), the carrier of VarPoset holds ( b2 = vars-function C iff for T being quasi-type of C holds b2 . T = vars T );
d53_abcmiz_1:: for L being non empty Poset holds ( L is smooth iff ex C being initialized ConstructorSignature ex f being Function of L,VarPoset st ( the carrier of L c= QuasiTypes C & f = (vars-function C) | the carrier of L & ( for x, y being Element of L holds f preserves_sup_of {x,y} ) ) );
d54_abcmiz_1:: for S being ManySortedSign holds ( S is with_an_operation_for_each_sort iff the carrier of S c= rng the ResultSort of S );
d55_abcmiz_1:: for S being ManySortedSign for X being ManySortedSet of the carrier of S holds ( X is with_missing_variables iff X " {{}} c= rng the ResultSort of S );
d56_abcmiz_1:: for S being non void Signature for X being ManySortedSet of the carrier of S for b3 being UnOp of (Union the Sorts of (Free (S,X))) holds ( b3 is term-transformation of S,X iff for s being SortSymbol of S holds b3 .: ( the Sorts of (Free (S,X)) . s) c= the Sorts of (Free (S,X)) . s );
d57_abcmiz_1:: for S being non void Signature for X being ManySortedSet of the carrier of S for t being term-transformation of S,X holds ( t is substitution iff for o being OperSymbol of S for p, q being FinSequence of (Free (S,X)) st [o, the carrier of S] -tree p in Union the Sorts of (Free (S,X)) & q = t * p holds t . ([o, the carrier of S] -tree p) = [o, the carrier of S] -tree q );
d58_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C holds ( f is irrelevant iff for x being variable st x in dom f holds ex y being variable st f . x = y -term C );
d59_abcmiz_1:: for C being initialized ConstructorSignature for X being Subset of Vars holds C idval X = { [x,(x -term C)] where x is variable : x in X } ;
d6_abcmiz_1:: a_Term = 2;
d60_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for b3 being term-transformation of C, MSVars C holds ( b3 = f # iff ( ( for x being variable holds ( ( x in dom f implies b3 . (x -term C) = f . x ) & ( not x in dom f implies b3 . (x -term C) = x -term C ) ) ) & ( for c being constructor OperSymbol of C for p, q being FinSequence of QuasiTerms C st len p = len (the_arity_of c) & q = b3 * p holds b3 . (c -trm p) = c -trm q ) & ( for a being expression of C, an_Adj C holds b3 . ((non_op C) term a) = (non_op C) term (b3 . a) ) & ( for a being expression of C, an_Adj C for t being expression of C, a_Type C holds b3 . ((ast C) term (a,t)) = (ast C) term ((b3 . a),(b3 . t)) ) ) );
d61_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for e being expression of C holds e at f = (f #) . e;
d62_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for p being FinSequence st rng p c= Union the Sorts of (Free (C,(MSVars C))) holds p at f = (f #) * p;
d63_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for p being FinSequence of QuasiTerms C holds p at f = (f #) * p;
d64_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for A being Subset of (QuasiAdjs C) holds A at f = { (a at f) where a is quasi-adjective of C : a in A } ;
d65_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for T being quasi-type of C holds T at f = ((adjs T) at f) ast ((the_base_of T) at f);
d66_abcmiz_1:: for C being initialized ConstructorSignature for f, g, b4 being valuation of C holds ( b4 = f at g iff ( dom b4 = (dom f) \/ (dom g) & ( for x being variable st x in dom b4 holds b4 . x = ((x -term C) at f) at g ) ) );
d7_abcmiz_1:: * = 0 ;
d8_abcmiz_1:: non_op = 1;
d9_abcmiz_1:: for C being Signature holds ( C is constructor iff ( the carrier of C = {a_Type,an_Adj,a_Term} & {*,non_op} c= the carrier' of C & the Arity of C . * = <*an_Adj,a_Type*> & the Arity of C . non_op = <*an_Adj*> & the ResultSort of C . * = a_Type & the ResultSort of C . non_op = an_Adj & ( for o being Element of the carrier' of C st o <> * & o <> non_op holds the Arity of C . o in {a_Term} * ) ) );
s1_abcmiz_1:: scheme Sch14{ F1() -> set , F2( set ) -> set , P1[ set ] } : varcl (union { F2(z) where z is Element of F1() : P1[z] } ) = union { (varcl F2(z)) where z is Element of F1() : P1[z] }
s2_abcmiz_1:: scheme StructInd{ F1() -> initialized ConstructorSignature, P1[ set ], F2() -> expression of F1() } : P1[F2()] provided A1: for x being variable holds P1[x -term F1()] and A2: for c being constructor OperSymbol of F1() for p being FinSequence of QuasiTerms F1() st len p = len (the_arity_of c) & ( for t being quasi-term of F1() st t in rng p holds P1[t] ) holds P1[c -trm p] and A3: for a being expression of F1(), an_Adj F1() st P1[a] holds P1[(non_op F1()) term a] and A4: for a being expression of F1(), an_Adj F1() st P1[a] holds for t being expression of F1(), a_Type F1() st P1[t] holds P1[(ast F1()) term (a,t)]
s3_abcmiz_1:: scheme StructDef{ F1() -> initialized ConstructorSignature, F2( set ) -> expression of F1(), F3( set ) -> expression of F1(), F4( set , set ) -> expression of F1(), F5( set , set ) -> expression of F1() } : ex f being term-transformation of F1(), MSVars F1() st ( ( for x being variable holds f . (x -term F1()) = F2(x) ) & ( for c being constructor OperSymbol of F1() for p, q being FinSequence of QuasiTerms F1() st len p = len (the_arity_of c) & q = f * p holds f . (c -trm p) = F4(c,q) ) & ( for a being expression of F1(), an_Adj F1() holds f . ((non_op F1()) term a) = F3((f . a)) ) & ( for a being expression of F1(), an_Adj F1() for t being expression of F1(), a_Type F1() holds f . ((ast F1()) term (a,t)) = F5((f . a),(f . t)) ) ) provided A1: for x being variable holds F2(x) is quasi-term of F1() and A2: for c being constructor OperSymbol of F1() for p being FinSequence of QuasiTerms F1() st len p = len (the_arity_of c) holds F4(c,p) is expression of F1(), the_result_sort_of c and A3: for a being expression of F1(), an_Adj F1() holds F3(a) is expression of F1(), an_Adj F1() and A4: for a being expression of F1(), an_Adj F1() for t being expression of F1(), a_Type F1() holds F5(a,t) is expression of F1(), a_Type F1()
t1_abcmiz_1:: for x being set for f being Function holds f . x c= Union f
t10_abcmiz_1:: for A being set holds varcl (union A) = union { (varcl a) where a is Element of A : verum }
t100_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds vars (Non a) = vars a by Th99;
t101_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C holds varcl (vars T) = vars T ;
t102_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C for a being quasi-adjective of C holds variables_in (a ast T) = (variables_in a) \/ (variables_in T)
t103_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C for a being quasi-adjective of C holds vars (a ast T) = (vars a) \/ (vars T)
t104_abcmiz_1:: for C being initialized ConstructorSignature for q being pure expression of C, a_Type C for A being finite Subset of (QuasiAdjs C) holds variables_in (A ast q) = (union { (variables_in a) where a is quasi-adjective of C : a in A } ) \/ (variables_in q)
t105_abcmiz_1:: for C being initialized ConstructorSignature for q being pure expression of C, a_Type C for A being finite Subset of (QuasiAdjs C) holds vars (A ast q) = (union { (vars a) where a is quasi-adjective of C : a in A } ) \/ (vars q)
t106_abcmiz_1:: for C being initialized ConstructorSignature for q being pure expression of C, a_Type C holds variables_in (({} (QuasiAdjs C)) ast q) = variables_in q
t107_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C holds ( e is ground iff variables_in e = {} )
t108_abcmiz_1:: for C being initialized ConstructorSignature for t being ground pure expression of C, a_Type C holds ({} (QuasiAdjs C)) ast t is ground
t109_abcmiz_1:: for x, y being Element of VarPoset holds ( x <= y iff y c= x )
t11_abcmiz_1:: for X, Y being set holds varcl (X \/ Y) = (varcl X) \/ (varcl Y)
t110_abcmiz_1:: for x being set holds ( x is Element of VarPoset iff ( x is finite Subset of Vars & varcl x = x ) )
t111_abcmiz_1:: for V1, V2 being Element of VarPoset holds ( V1 "\/" V2 = V1 /\ V2 & V1 "/\" V2 = V1 \/ V2 )
t112_abcmiz_1:: for X being non empty Subset of VarPoset holds ( ex_sup_of X, VarPoset & sup X = meet X )
t113_abcmiz_1:: Top VarPoset = {}
t114_abcmiz_1:: for S being non void Signature for X being ManySortedSet of the carrier of S holds ( X is with_missing_variables iff for s being SortSymbol of S st X . s = {} holds ex o being OperSymbol of S st the_result_sort_of o = s )
t115_abcmiz_1:: for D1, D2 being non empty DTConstrStr st the Rules of D1 c= the Rules of D2 holds ( NonTerminals D1 c= NonTerminals D2 & the carrier of D1 /\ (Terminals D2) c= Terminals D1 & ( Terminals D1 c= Terminals D2 implies the carrier of D1 c= the carrier of D2 ) )
t116_abcmiz_1:: for D1, D2 being non empty DTConstrStr st Terminals D1 c= Terminals D2 & the Rules of D1 c= the Rules of D2 holds TS D1 c= TS D2
t117_abcmiz_1:: for S being ManySortedSign for X, Y being ManySortedSet of the carrier of S st X c= Y & X is with_missing_variables holds Y is with_missing_variables
t118_abcmiz_1:: for S being set for X, Y being ManySortedSet of S st X c= Y holds Union (coprod X) c= Union (coprod Y)
t119_abcmiz_1:: for S being non void Signature for X, Y being ManySortedSet of the carrier of S st X c= Y holds the carrier of (DTConMSA X) c= the carrier of (DTConMSA Y) by Th118, XBOOLE_1:9;
t12_abcmiz_1:: for A being non empty set st ( for a being Element of A holds varcl a = a ) holds varcl (meet A) = meet A
t120_abcmiz_1:: for S being non void Signature for X being ManySortedSet of the carrier of S st X is with_missing_variables holds ( NonTerminals (DTConMSA X) = [: the carrier' of S,{ the carrier of S}:] & Terminals (DTConMSA X) = Union (coprod X) )
t121_abcmiz_1:: for S being non void Signature for X, Y being ManySortedSet of the carrier of S st X c= Y & X is with_missing_variables holds ( Terminals (DTConMSA X) c= Terminals (DTConMSA Y) & the Rules of (DTConMSA X) c= the Rules of (DTConMSA Y) & TS (DTConMSA X) c= TS (DTConMSA Y) )
t122_abcmiz_1:: for C being initialized ConstructorSignature for t being set holds ( t in Terminals (DTConMSA (MSVars C)) iff ex x being variable st t = [x,(a_Term C)] )
t123_abcmiz_1:: for C being initialized ConstructorSignature for t being set holds ( t in NonTerminals (DTConMSA (MSVars C)) iff ( t = [(ast C), the carrier of C] or t = [(non_op C), the carrier of C] or ex c being constructor OperSymbol of C st t = [c, the carrier of C] ) )
t124_abcmiz_1:: for S being non void Signature for X being with_missing_variables ManySortedSet of the carrier of S for t being set st t in Union the Sorts of (Free (S,X)) holds t is Term of S,(X \/ ( the carrier of S --> {0}))
t125_abcmiz_1:: for S being non void Signature for X being with_missing_variables ManySortedSet of the carrier of S for t being Term of S,(X \/ ( the carrier of S --> {0})) st t in Union the Sorts of (Free (S,X)) holds t in the Sorts of (Free (S,X)) . (the_sort_of t)
t126_abcmiz_1:: for G being non empty DTConstrStr for s being Element of G for p being FinSequence st s ==> p holds p is FinSequence of the carrier of G
t127_abcmiz_1:: for S being non void Signature for X, Y being ManySortedSet of the carrier of S for g1 being Symbol of (DTConMSA X) for g2 being Symbol of (DTConMSA Y) for p1 being FinSequence of the carrier of (DTConMSA X) for p2 being FinSequence of the carrier of (DTConMSA Y) st g1 = g2 & p1 = p2 & g1 ==> p1 holds g2 ==> p2
t128_abcmiz_1:: for S being non void Signature for X being with_missing_variables ManySortedSet of the carrier of S holds Union the Sorts of (Free (S,X)) = TS (DTConMSA X)
t129_abcmiz_1:: for S being non void Signature for X being non empty ManySortedSet of the carrier of S for f being UnOp of (Union the Sorts of (Free (S,X))) holds ( f is term-transformation of S,X iff for s being SortSymbol of S for a being set st a in the Sorts of (Free (S,X)) . s holds f . a in the Sorts of (Free (S,X)) . s )
t13_abcmiz_1:: for X, Y being set holds varcl ((varcl X) /\ (varcl Y)) = (varcl X) /\ (varcl Y)
t130_abcmiz_1:: for S being non void Signature for X being non empty ManySortedSet of the carrier of S for f being term-transformation of S,X for s being SortSymbol of S for p being FinSequence of the Sorts of (Free (S,X)) . s holds ( f * p is FinSequence of the Sorts of (Free (S,X)) . s & card (f * p) = len p )
t131_abcmiz_1:: for C being initialized ConstructorSignature for X being Subset of Vars holds ( dom (C idval X) = X & ( for x being variable st x in X holds (C idval X) . x = x -term C ) )
t132_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for x being variable st not x in dom f holds (x -term C) at f = x -term C by Def60;
t133_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for x being variable st x in dom f holds (x -term C) at f = f . x by Def60;
t134_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C for f being valuation of C st len p = len (the_arity_of c) holds (c -trm p) at f = c -trm (p at f) by Def60;
t135_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C for f being valuation of C holds ((non_op C) term a) at f = (non_op C) term (a at f) by Def60;
t136_abcmiz_1:: for C being initialized ConstructorSignature for t being expression of C, a_Type C for a being expression of C, an_Adj C for f being valuation of C holds ((ast C) term (a,t)) at f = (ast C) term ((a at f),(t at f)) by Def60;
t137_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C for X being Subset of Vars holds e at (C idval X) = e
t138_abcmiz_1:: for C being initialized ConstructorSignature for X being Subset of Vars holds (C idval X) # = id (Union the Sorts of (Free (C,(MSVars C))))
t139_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C for f being empty valuation of C holds e at f = e
t14_abcmiz_1:: for V being ManySortedSet of NAT st V . 0 = { [{},i] where i is Element of NAT : verum } & ( for n being Nat holds V . (n + 1) = { [(varcl A),j] where A is Subset of (V . n), j is Element of NAT : A is finite } ) holds for i, j being Element of NAT st i <= j holds V . i c= V . j
t140_abcmiz_1:: for C being initialized ConstructorSignature for f being empty valuation of C holds f # = id (Union the Sorts of (Free (C,(MSVars C))))
t141_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C for f being valuation of C holds (Non a) at f = Non (a at f)
t142_abcmiz_1:: for C being initialized ConstructorSignature for f being one-to-one irrelevant valuation of C ex g being one-to-one irrelevant valuation of C st for x, y being variable holds ( x in dom f & f . x = y -term C iff ( y in dom g & g . y = x -term C ) )
t143_abcmiz_1:: for C being initialized ConstructorSignature for f, g being one-to-one irrelevant valuation of C st ( for x, y being variable st x in dom f & f . x = y -term C holds ( y in dom g & g . y = x -term C ) ) holds for e being expression of C st variables_in e c= dom f holds (e at f) at g = e
t144_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for A being Subset of (QuasiAdjs C) for a being quasi-adjective of C st A = {a} holds A at f = {(a at f)}
t145_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for A, B being Subset of (QuasiAdjs C) holds (A \/ B) at f = (A at f) \/ (B at f)
t146_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for A, B being Subset of (QuasiAdjs C) st A c= B holds A at f c= B at f
t147_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for T being quasi-type of C holds ( adjs (T at f) = (adjs T) at f & the_base_of (T at f) = (the_base_of T) at f ) by MCART_1:7;
t148_abcmiz_1:: for C being initialized ConstructorSignature for f being valuation of C for T being quasi-type of C for a being quasi-adjective of C holds (a ast T) at f = (a at f) ast (T at f)
t149_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C for f1, f2 being valuation of C holds (e at f1) at f2 = e at (f1 at f2)
t15_abcmiz_1:: for V being ManySortedSet of NAT st V . 0 = { [{},i] where i is Element of NAT : verum } & ( for n being Nat holds V . (n + 1) = { [(varcl A),j] where A is Subset of (V . n), j is Element of NAT : A is finite } ) holds for A being finite Subset of Vars ex i being Element of NAT st A c= V . i
t150_abcmiz_1:: for C being initialized ConstructorSignature for A being Subset of (QuasiAdjs C) for f1, f2 being valuation of C holds (A at f1) at f2 = A at (f1 at f2)
t151_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C for f1, f2 being valuation of C holds (T at f1) at f2 = T at (f1 at f2)
t16_abcmiz_1:: { [{},i] where i is Element of NAT : verum } c= Vars
t17_abcmiz_1:: for A being finite Subset of Vars for i being Nat holds [(varcl A),i] in Vars
t18_abcmiz_1:: Vars = { [(varcl A),j] where A is Subset of Vars, j is Element of NAT : A is finite }
t19_abcmiz_1:: varcl Vars = Vars
t2_abcmiz_1:: for x being set for f being Function st Union f = {} holds f . x = {} by Th1, XBOOLE_1:3;
t20_abcmiz_1:: for X being set st the_rank_of X is finite holds X is finite
t21_abcmiz_1:: for X being set holds the_rank_of (varcl X) = the_rank_of X
t22_abcmiz_1:: for X being finite Subset of (Rank omega) holds X in Rank omega
t23_abcmiz_1:: Vars c= Rank omega
t24_abcmiz_1:: for A being finite Subset of Vars holds varcl A is finite Subset of Vars
t25_abcmiz_1:: for i being Nat holds [{},i] in Vars
t26_abcmiz_1:: for j being Element of NAT for A being Subset of Vars holds varcl {[(varcl A),j]} = (varcl A) \/ {[(varcl A),j]}
t27_abcmiz_1:: for x being variable holds varcl {x} = (vars x) \/ {x}
t28_abcmiz_1:: for i being Nat for x being variable holds [((vars x) \/ {x}),i] in Vars
t29_abcmiz_1:: <*> Vars in QuasiLoci
t3_abcmiz_1:: for f being Function for x, y being set st f = [x,y] holds x = y
t30_abcmiz_1:: for l being one-to-one FinSequence of Vars holds ( l is quasi-loci iff for i being Nat for x being variable st i in dom l & x = l . i holds for y being variable st y in vars x holds ex j being Nat st ( j in dom l & j < i & y = l . j ) )
t31_abcmiz_1:: for l being quasi-loci for x being variable holds ( l ^ <*x*> is quasi-loci iff ( not x in rng l & vars x c= rng l ) )
t32_abcmiz_1:: for p, q being FinSequence st p ^ q is quasi-loci holds ( p is quasi-loci & q is FinSequence of Vars )
t33_abcmiz_1:: for l being quasi-loci holds varcl (rng l) = rng l
t34_abcmiz_1:: for x being variable holds ( <*x*> is quasi-loci iff vars x = {} )
t35_abcmiz_1:: for x, y being variable holds ( <*x,y*> is quasi-loci iff ( vars x = {} & x <> y & vars y c= {x} ) )
t36_abcmiz_1:: for x, y, z being variable holds ( <*x,y,z*> is quasi-loci iff ( vars x = {} & x <> y & vars y c= {x} & x <> z & y <> z & vars z c= {x,y} ) )
t37_abcmiz_1:: for S being ConstructorSignature for o being OperSymbol of S st o is constructor holds the_arity_of o = (len (the_arity_of o)) |-> a_Term
t38_abcmiz_1:: for C being ConstructorSignature holds ( the_arity_of (non_op C) = <*(an_Adj C)*> & the_result_sort_of (non_op C) = an_Adj C & the_arity_of (ast C) = <*(an_Adj C),(a_Type C)*> & the_result_sort_of (ast C) = a_Type C ) by Def9;
t39_abcmiz_1:: {*,non_op} misses Constructors
t4_abcmiz_1:: for X, Y being set holds (id X) .: Y c= Y
t40_abcmiz_1:: for c being Element of Constructors holds ( ( kind_of c = a_Type implies c in Modes ) & ( c in Modes implies kind_of c = a_Type ) & ( kind_of c = an_Adj implies c in Attrs ) & ( c in Attrs implies kind_of c = an_Adj ) & ( kind_of c = a_Term implies c in Funcs ) & ( c in Funcs implies kind_of c = a_Term ) )
t41_abcmiz_1:: for z being set for C being initialized ConstructorSignature for s being SortSymbol of C holds ( z is expression of C,s iff z in the Sorts of (Free (C,(MSVars C))) . s )
t42_abcmiz_1:: for C being initialized ConstructorSignature for o being OperSymbol of C st len (the_arity_of o) = 1 holds for a being expression of C st ex s being SortSymbol of C st ( s = (the_arity_of o) . 1 & a is expression of C,s ) holds [o, the carrier of C] -tree <*a*> is expression of C, the_result_sort_of o
t43_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds ( (non_op C) term a is expression of C, an_Adj C & (non_op C) term a = [non_op, the carrier of C] -tree <*a*> )
t44_abcmiz_1:: for C being initialized ConstructorSignature for a, b being expression of C, an_Adj C st (non_op C) term a = (non_op C) term b holds a = b
t45_abcmiz_1:: for C being initialized ConstructorSignature for o being OperSymbol of C st len (the_arity_of o) = 2 holds for a, b being expression of C st ex s1, s2 being SortSymbol of C st ( s1 = (the_arity_of o) . 1 & s2 = (the_arity_of o) . 2 & a is expression of C,s1 & b is expression of C,s2 ) holds [o, the carrier of C] -tree <*a,b*> is expression of C, the_result_sort_of o
t46_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C for t being expression of C, a_Type C holds ( (ast C) term (a,t) is expression of C, a_Type C & (ast C) term (a,t) = [*, the carrier of C] -tree <*a,t*> )
t47_abcmiz_1:: for C being initialized ConstructorSignature for a, b being expression of C, an_Adj C for t1, t2 being expression of C, a_Type C st (ast C) term (a,t1) = (ast C) term (b,t2) holds ( a = b & t1 = t2 )
t48_abcmiz_1:: for C being initialized ConstructorSignature for s1, s2 being SortSymbol of C st s1 <> s2 holds for t1 being expression of C,s1 for t2 being expression of C,s2 holds t1 <> t2
t49_abcmiz_1:: for z being set for C being initialized ConstructorSignature holds ( z is quasi-term of C iff z in QuasiTerms C ) by Th41;
t5_abcmiz_1:: for S being non void Signature for X being V8() ManySortedSet of the carrier of S for t being Term of S,X holds not t is pair
t50_abcmiz_1:: for x1, x2 being variable for C1, C2 being initialized ConstructorSignature st x1 -term C1 = x2 -term C2 holds x1 = x2
t51_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for p being DTree-yielding FinSequence holds ( [c, the carrier of C] -tree p is expression of C iff ( len p = len (the_arity_of c) & p in (QuasiTerms C) * ) )
t52_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len (the_arity_of c) holds c -trm p is expression of C, the_result_sort_of c
t53_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C holds ( ex x being variable st e = x -term C or ex c being constructor OperSymbol of C ex p being FinSequence of QuasiTerms C st ( len p = len (the_arity_of c) & e = c -trm p ) or ex a being expression of C, an_Adj C st e = (non_op C) term a or ex a being expression of C, an_Adj C ex t being expression of C, a_Type C st e = (ast C) term (a,t) )
t54_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for a being expression of C, an_Adj C for p being FinSequence of QuasiTerms C st len p = len (the_arity_of c) holds c -trm p <> (non_op C) term a
t55_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for a being expression of C, an_Adj C for t being expression of C, a_Type C for p being FinSequence of QuasiTerms C st len p = len (the_arity_of c) holds c -trm p <> (ast C) term (a,t)
t56_abcmiz_1:: for C being initialized ConstructorSignature for a, b being expression of C, an_Adj C for t being expression of C, a_Type C holds (non_op C) term a <> (ast C) term (b,t)
t57_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C st e . {} = [non_op, the carrier of C] holds ex a being expression of C, an_Adj C st e = (non_op C) term a
t58_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C st e . {} = [*, the carrier of C] holds ex a being expression of C, an_Adj C ex t being expression of C, a_Type C st e = (ast C) term (a,t)
t59_abcmiz_1:: for C being initialized ConstructorSignature for a being positive expression of C, an_Adj C holds Non a = (non_op C) term a
t6_abcmiz_1:: for x, y, z being set st x in {z} * & y in {z} * & card x = card y holds x = y
t60_abcmiz_1:: for C being initialized ConstructorSignature for a being non positive expression of C, an_Adj C ex a9 being expression of C, an_Adj C st ( a = (non_op C) term a9 & Non a = a9 )
t61_abcmiz_1:: for C being initialized ConstructorSignature for a being negative expression of C, an_Adj C ex a9 being positive expression of C, an_Adj C st ( a = (non_op C) term a9 & Non a = a9 )
t62_abcmiz_1:: for C being initialized ConstructorSignature for a being non positive expression of C, an_Adj C holds (non_op C) term (Non a) = a
t63_abcmiz_1:: for z being set for C being initialized ConstructorSignature holds ( z is quasi-adjective of C iff z in QuasiAdjs C )
t64_abcmiz_1:: for z being set for C being initialized ConstructorSignature holds ( z is quasi-adjective of C iff ( z is positive expression of C, an_Adj C or z is negative expression of C, an_Adj C ) ) by Def39;
t65_abcmiz_1:: for C being initialized ConstructorSignature for a being positive quasi-adjective of C ex v being constructor OperSymbol of C st ( the_result_sort_of v = an_Adj C & ex p being FinSequence of QuasiTerms C st ( len p = len (the_arity_of v) & a = v -trm p ) )
t66_abcmiz_1:: for C being initialized ConstructorSignature for p being FinSequence of QuasiTerms C for v being constructor OperSymbol of C st the_result_sort_of v = an_Adj C & len p = len (the_arity_of v) holds v -trm p is positive quasi-adjective of C
t67_abcmiz_1:: for C being initialized ConstructorSignature for a being quasi-adjective of C holds Non (Non a) = a
t68_abcmiz_1:: for C being initialized ConstructorSignature for a1, a2 being quasi-adjective of C st Non a1 = Non a2 holds a1 = a2
t69_abcmiz_1:: for C being initialized ConstructorSignature for a being quasi-adjective of C holds Non a <> a
t7_abcmiz_1:: for S being non void Signature for X being V9() ManySortedSet of the carrier of S for t being Element of (Free (S,X)) holds ( ex s being SortSymbol of S ex v being set st ( t = root-tree [v,s] & v in X . s ) or ex o being OperSymbol of S ex p being FinSequence of (Free (S,X)) st ( t = [o, the carrier of S] -tree p & len p = len (the_arity_of o) & p is DTree-yielding & p is ArgumentSeq of Sym (o,(X \/ ( the carrier of S --> {0}))) ) )
t70_abcmiz_1:: for C being initialized ConstructorSignature for m being OperSymbol of C st the_result_sort_of m = a_Type & the_arity_of m = {} holds ex t being expression of C, a_Type C st ( t = root-tree [m, the carrier of C] & t is pure )
t71_abcmiz_1:: for C being initialized ConstructorSignature for v being OperSymbol of C st the_result_sort_of v = an_Adj & the_arity_of v = {} holds ex a being expression of C, an_Adj C st ( a = root-tree [v, the carrier of C] & a is positive )
t72_abcmiz_1:: for z being set for C being initialized ConstructorSignature holds ( z is quasi-type of C iff ex A being finite Subset of (QuasiAdjs C) ex q being pure expression of C, a_Type C st z = [A,q] )
t73_abcmiz_1:: for x, y being set for C being initialized ConstructorSignature holds ( [x,y] is quasi-type of C iff ( x is finite Subset of (QuasiAdjs C) & y is pure expression of C, a_Type C ) )
t74_abcmiz_1:: for C being initialized ConstructorSignature for q being pure expression of C, a_Type C ex m being constructor OperSymbol of C st ( the_result_sort_of m = a_Type C & ex p being FinSequence of QuasiTerms C st ( len p = len (the_arity_of m) & q = m -trm p ) )
t75_abcmiz_1:: for C being initialized ConstructorSignature for p being FinSequence of QuasiTerms C for m being constructor OperSymbol of C st the_result_sort_of m = a_Type C & len p = len (the_arity_of m) holds m -trm p is pure expression of C, a_Type C
t76_abcmiz_1:: for C being initialized ConstructorSignature holds ( QuasiTerms C misses QuasiAdjs C & QuasiTerms C misses QuasiTypes C & QuasiTypes C misses QuasiAdjs C )
t77_abcmiz_1:: for C being initialized ConstructorSignature for e being set holds ( ( e is quasi-term of C implies not e is quasi-adjective of C ) & ( e is quasi-term of C implies not e is quasi-type of C ) & ( e is quasi-type of C implies not e is quasi-adjective of C ) ) by Th48;
t78_abcmiz_1:: for C being initialized ConstructorSignature for q being pure expression of C, a_Type C for A being finite Subset of (QuasiAdjs C) holds ( adjs (A ast q) = A & the_base_of (A ast q) = q ) ;
t79_abcmiz_1:: for C being initialized ConstructorSignature for A1, A2 being finite Subset of (QuasiAdjs C) for q1, q2 being pure expression of C, a_Type C st A1 ast q1 = A2 ast q2 holds ( A1 = A2 & q1 = q2 ) by XTUPLE_0:1;
t8_abcmiz_1:: varcl {} = {}
t80_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C holds T = (adjs T) ast (the_base_of T)
t81_abcmiz_1:: for C being initialized ConstructorSignature for T1, T2 being quasi-type of C st adjs T1 = adjs T2 & the_base_of T1 = the_base_of T2 holds T1 = T2
t82_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C for a being quasi-adjective of C holds ( adjs (a ast T) = {a} \/ (adjs T) & the_base_of (a ast T) = the_base_of T ) by MCART_1:7;
t83_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C for a being quasi-adjective of C holds a ast (a ast T) = a ast T
t84_abcmiz_1:: for C being initialized ConstructorSignature for T being quasi-type of C for a, b being quasi-adjective of C holds a ast (b ast T) = b ast (a ast T)
t85_abcmiz_1:: for C being initialized ConstructorSignature for e being expression of C holds varcl (vars e) = vars e ;
t86_abcmiz_1:: for C being initialized ConstructorSignature for x being variable holds variables_in (x -term C) = {x} by MSAFREE3:10;
t87_abcmiz_1:: for C being initialized ConstructorSignature for x being variable holds vars (x -term C) = {x} \/ (vars x)
t88_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for e being expression of C for p being DTree-yielding FinSequence st e = [c, the carrier of C] -tree p holds variables_in e = union { (variables_in t) where t is quasi-term of C : t in rng p }
t89_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for e being expression of C for p being DTree-yielding FinSequence st e = [c, the carrier of C] -tree p holds vars e = union { (vars t) where t is quasi-term of C : t in rng p }
t9_abcmiz_1:: for A, B being set st A c= B holds varcl A c= varcl B
t90_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len (the_arity_of c) holds variables_in (c -trm p) = union { (variables_in t) where t is quasi-term of C : t in rng p }
t91_abcmiz_1:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len (the_arity_of c) holds vars (c -trm p) = union { (vars t) where t is quasi-term of C : t in rng p }
t92_abcmiz_1:: for S being ManySortedSign for o being set holds S variables_in ([o, the carrier of S] -tree {}) = [[0]] the carrier of S
t93_abcmiz_1:: for S being ManySortedSign for o being set for t being DecoratedTree holds S variables_in ([o, the carrier of S] -tree <*t*>) = S variables_in t
t94_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds variables_in ((non_op C) term a) = variables_in a
t95_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds vars ((non_op C) term a) = vars a by Th94;
t96_abcmiz_1:: for S being ManySortedSign for o being set for t1, t2 being DecoratedTree holds S variables_in ([o, the carrier of S] -tree <*t1,t2*>) = (S variables_in t1) \/ (S variables_in t2)
t97_abcmiz_1:: for C being initialized ConstructorSignature for t being expression of C, a_Type C for a being expression of C, an_Adj C holds variables_in ((ast C) term (a,t)) = (variables_in a) \/ (variables_in t)
t98_abcmiz_1:: for C being initialized ConstructorSignature for t being expression of C, a_Type C for a being expression of C, an_Adj C holds vars ((ast C) term (a,t)) = (vars a) \/ (vars t)
t99_abcmiz_1:: for C being initialized ConstructorSignature for a being expression of C, an_Adj C holds variables_in (Non a) = variables_in a
d1_abcmiz_a:: for C being ConstructorSignature holds ( C is standardized iff for o being OperSymbol of C st o is constructor holds ( o in Constructors & o `1 = the_result_sort_of o & card ((o `2) `1) = len (the_arity_of o) ) );
d10_abcmiz_a:: for C being initialized ConstructorSignature for e being expression of C for b3 being FinSequence of (Free (C,(MSVars C))) holds ( b3 = args e iff e = (e . {}) -tree b3 );
d11_abcmiz_a:: for C being initialized ConstructorSignature for e being expression of C holds ( e is constructor iff ( e is compound & main-constr e is constructor OperSymbol of C ) );
d12_abcmiz_a:: for C being non void Signature holds ( C is arity-rich iff for n being Nat for s being SortSymbol of C holds { o where o is OperSymbol of C : ( the_result_sort_of o = s & len (the_arity_of o) = n ) } is infinite );
d13_abcmiz_a:: for C being non void Signature for o being OperSymbol of C holds ( o is nullary iff the_arity_of o = {} );
d14_abcmiz_a:: for C being non void Signature for o being OperSymbol of C holds ( o is unary iff len (the_arity_of o) = 1 );
d15_abcmiz_a:: for C being non void Signature for o being OperSymbol of C holds ( o is binary iff len (the_arity_of o) = 2 );
d16_abcmiz_a:: for c being Element of Constructors holds @ c = c;
d17_abcmiz_a:: set-type = ({} (QuasiAdjs MaxConstrSign)) ast ((@ set-constr) term);
d18_abcmiz_a:: for l being FinSequence of Vars for b2 being FinSequence of QuasiTerms MaxConstrSign holds ( b2 = args l iff ( len b2 = len l & ( for i being Nat st i in dom l holds b2 . i = (l /. i) -term MaxConstrSign ) ) );
d19_abcmiz_a:: for c being Element of Constructors holds base_exp_of c = (@ c) -trm (args (loci_of c));
d2_abcmiz_a:: for C being initialized standardized ConstructorSignature for c being constructor OperSymbol of C holds loci_of c = (c `2) `1 ;
d20_abcmiz_a:: for T being quasi-type holds constrs T = (constrs (the_base_of T)) \/ (union { (constrs a) where a is quasi-adjective : a in adjs T } );
d21_abcmiz_a:: for C being initialized ConstructorSignature for t, p being expression of C holds ( t matches_with p iff ex f being valuation of C st t = p at f );
d22_abcmiz_a:: for C being initialized ConstructorSignature for A, B being Subset of (QuasiAdjs C) holds ( A matches_with B iff ex f being valuation of C st B at f c= A );
d23_abcmiz_a:: for C being initialized ConstructorSignature for T, P being quasi-type of C holds ( T matches_with P iff ex f being valuation of C st ( (adjs P) at f c= adjs T & (the_base_of P) at f = the_base_of T ) );
d24_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2 being expression of C for f being valuation of C holds ( f unifies t1,t2 iff t1 at f = t2 at f );
d25_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2 being expression of C holds ( t1,t2 are_unifiable iff ex f being valuation of C st f unifies t1,t2 );
d26_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2 being expression of C holds ( t1,t2 are_weakly-unifiable iff ex g being one-to-one irrelevant valuation of C st ( variables_in t2 c= dom g & t1,t2 at g are_unifiable ) );
d27_abcmiz_a:: for C being initialized ConstructorSignature for t, t1, t2 being expression of C holds ( t is_a_unification_of t1,t2 iff ex f being valuation of C st ( f unifies t1,t2 & t = t1 at f ) );
d28_abcmiz_a:: for C being initialized ConstructorSignature for t, t1, t2 being expression of C holds ( t is_a_general-unification_of t1,t2 iff ( t is_a_unification_of t1,t2 & ( for u being expression of C st u is_a_unification_of t1,t2 holds u matches_with t ) ) );
d29_abcmiz_a:: for d being PartFunc of Vars,QuasiTypes holds ( d is even iff for x being variable for T being quasi-type st x in dom d & T = d . x holds vars T = vars x );
d3_abcmiz_a:: QuasiAdjs = QuasiAdjs MaxConstrSign;
d30_abcmiz_a:: for l being quasi-loci for b2 being PartFunc of Vars,QuasiTypes holds ( b2 is type-distribution of l iff ( dom b2 = rng l & b2 is even ) );
d4_abcmiz_a:: QuasiTerms = QuasiTerms MaxConstrSign;
d5_abcmiz_a:: QuasiTypes = QuasiTypes MaxConstrSign;
d6_abcmiz_a:: set-constr = [a_Type,[{},0]];
d7_abcmiz_a:: for C being initialized ConstructorSignature for e, b3 being expression of C holds ( b3 is subexpression of e iff b3 in Subtrees e );
d8_abcmiz_a:: for C being initialized ConstructorSignature for e being expression of C holds constrs e = (proj1 (rng e)) /\ { o where o is constructor OperSymbol of C : verum } ;
d9_abcmiz_a:: for C being initialized ConstructorSignature for e being expression of C holds ( ( e is compound implies main-constr e = (e . {}) `1 ) & ( not e is compound implies main-constr e = {} ) );
s1_abcmiz_a:: scheme MinimalElement{ F1() -> non empty finite set , P1[ set , set ] } : ex x being set st ( x in F1() & ( for y being set st y in F1() holds not P1[y,x] ) ) provided A1: for x, y being set st x in F1() & y in F1() & P1[x,y] holds not P1[y,x] and A2: for x, y, z being set st x in F1() & y in F1() & z in F1() & P1[x,y] & P1[y,z] holds P1[x,z]
s2_abcmiz_a:: scheme FiniteC{ F1() -> finite set , P1[ set ] } : P1[F1()] provided A1: for A being Subset of F1() st ( for B being set st B c< A holds P1[B] ) holds P1[A]
s3_abcmiz_a:: scheme Numeration{ F1() -> finite set , P1[ set , set ] } : ex s being one-to-one FinSequence st ( rng s = F1() & ( for i, j being Nat st i in dom s & j in dom s & P1[s . i,s . j] holds i < j ) ) provided A1: for x, y being set st x in F1() & y in F1() & P1[x,y] holds not P1[y,x] and A2: for x, y, z being set st x in F1() & y in F1() & z in F1() & P1[x,y] & P1[y,z] holds P1[x,z]
t1_abcmiz_a:: for x being pair set holds x = [(x `1),(x `2)] ;
t10_abcmiz_a:: for C being standardized ConstructorSignature holds Vars misses the carrier' of C
t11_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C holds ( ex x being Element of Vars st ( e = x -term C & e . {} = [x,a_Term] ) or ex o being OperSymbol of C st ( e . {} = [o, the carrier of C] & ( o in Constructors or o = * or o = non_op ) ) )
t12_abcmiz_a:: for C being initialized ConstructorSignature for e being expression of C for o being OperSymbol of C st e . {} = [o, the carrier of C] holds e is expression of C, the_result_sort_of o
t13_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C holds ( ( (e . {}) `1 = * implies e is expression of C, a_Type C ) & ( (e . {}) `1 = non_op implies e is expression of C, an_Adj C ) )
t14_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C holds ( ( (e . {}) `1 in Vars & (e . {}) `2 = a_Term & e is quasi-term of C ) or ( (e . {}) `2 = the carrier of C & ( ( (e . {}) `1 in Constructors & (e . {}) `1 in the carrier' of C ) or (e . {}) `1 = * or (e . {}) `1 = non_op ) ) )
t15_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C st (e . {}) `1 in Constructors holds e in the Sorts of (Free (C,(MSVars C))) . (((e . {}) `1) `1)
t16_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C holds ( not (e . {}) `1 in Vars iff (e . {}) `1 is OperSymbol of C )
t17_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C st (e . {}) `1 in Vars holds ex x being Element of Vars st ( x = (e . {}) `1 & e = x -term C )
t18_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C st (e . {}) `1 = * holds ex a being expression of C, an_Adj C ex q being expression of C, a_Type C st e = [*,3] -tree (a,q)
t19_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C st (e . {}) `1 = non_op holds ex a being expression of C, an_Adj C st e = [non_op,3] -tree a
t2_abcmiz_a:: for x being variable holds varcl (vars x) = vars x
t20_abcmiz_a:: for C being initialized standardized ConstructorSignature for e being expression of C st (e . {}) `1 in Constructors holds ex o being OperSymbol of C st ( o = (e . {}) `1 & the_result_sort_of o = o `1 & e is expression of C, the_result_sort_of o )
t21_abcmiz_a:: for C being initialized standardized ConstructorSignature for t being quasi-term of C holds ( t is compound iff ( (t . {}) `1 in Constructors & ((t . {}) `1) `1 = a_Term ) )
t22_abcmiz_a:: for C being initialized standardized ConstructorSignature for t being expression of C holds ( t is non compound quasi-term of C iff (t . {}) `1 in Vars )
t23_abcmiz_a:: for C being initialized standardized ConstructorSignature for t being expression of C holds ( t is quasi-term of C iff ( ( (t . {}) `1 in Constructors & ((t . {}) `1) `1 = a_Term ) or (t . {}) `1 in Vars ) )
t24_abcmiz_a:: for C being initialized standardized ConstructorSignature for a being expression of C holds ( a is positive quasi-adjective of C iff ( (a . {}) `1 in Constructors & ((a . {}) `1) `1 = an_Adj ) )
t25_abcmiz_a:: for C being initialized standardized ConstructorSignature for a being quasi-adjective of C holds ( a is negative iff (a . {}) `1 = non_op )
t26_abcmiz_a:: for C being initialized standardized ConstructorSignature for t being expression of C holds ( t is pure expression of C, a_Type C iff ( (t . {}) `1 in Constructors & ((t . {}) `1) `1 = a_Type ) )
t27_abcmiz_a:: ( kind_of set-constr = a_Type & loci_of set-constr = {} & index_of set-constr = 0 )
t28_abcmiz_a:: Constructors = [:{a_Type,an_Adj,a_Term},[:QuasiLoci,NAT:]:]
t29_abcmiz_a:: for i being Nat for l being quasi-loci holds ( [(rng l),i] in Vars & l ^ <*[(rng l),i]*> is quasi-loci )
t3_abcmiz_a:: for C being initialized ConstructorSignature for e being expression of C holds ( e is compound iff for x being Element of Vars holds not e = x -term C )
t30_abcmiz_a:: for i being Nat ex l being quasi-loci st len l = i
t31_abcmiz_a:: for X being finite Subset of Vars ex l being quasi-loci st rng l = varcl X
t32_abcmiz_a:: for X, o being set for p being DTree-yielding FinSequence st ex C being initialized ConstructorSignature st X = Union the Sorts of (Free (C,(MSVars C))) & o -tree p in X holds p is FinSequence of X
t33_abcmiz_a:: for C being initialized ConstructorSignature for e being expression of C holds e is subexpression of e
t34_abcmiz_a:: for x being variable for C being initialized ConstructorSignature holds main-constr (x -term C) = {} by Def9;
t35_abcmiz_a:: for C being initialized ConstructorSignature for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len (the_arity_of c) holds main-constr (c -trm p) = c
t36_abcmiz_a:: for C being initialized ConstructorSignature for e being constructor expression of C holds main-constr e in constrs e
t37_abcmiz_a:: for C being non void Signature for o being OperSymbol of C holds ( ( o is nullary implies not o is unary ) & ( o is nullary implies not o is binary ) & ( o is unary implies not o is binary ) )
t38_abcmiz_a:: for C being ConstructorSignature holds ( C is initialized iff ex m being OperSymbol of a_Type C ex a being OperSymbol of an_Adj C st ( m is nullary & a is nullary ) )
t39_abcmiz_a:: for C being initialized ConstructorSignature for o being nullary OperSymbol of C holds [o, the carrier of C] -tree {} is expression of C, the_result_sort_of o
t4_abcmiz_a:: for C being ConstructorSignature st C is standardized holds for o being OperSymbol of C holds ( o is constructor iff o in Constructors )
t40_abcmiz_a:: the_arity_of (@ set-constr) = {} by Def13;
t41_abcmiz_a:: ( adjs set-type = {} & the_base_of set-type = (@ set-constr) term ) by MCART_1:7;
t42_abcmiz_a:: for o being OperSymbol of MaxConstrSign holds ( o is constructor iff o in Constructors )
t43_abcmiz_a:: for m being nullary OperSymbol of MaxConstrSign holds main-constr (m term) = m
t44_abcmiz_a:: for m being constructor unary OperSymbol of MaxConstrSign for t being quasi-term holds main-constr (m term t) = m
t45_abcmiz_a:: for a being quasi-adjective holds main-constr ((non_op MaxConstrSign) term a) = non_op
t46_abcmiz_a:: for m being constructor binary OperSymbol of MaxConstrSign for t1, t2 being quasi-term holds main-constr (m term (t1,t2)) = m
t47_abcmiz_a:: for q being expression of MaxConstrSign , a_Type MaxConstrSign for a being quasi-adjective holds main-constr ((ast MaxConstrSign) term (a,q)) = *
t48_abcmiz_a:: for q being pure expression of MaxConstrSign , a_Type MaxConstrSign for A being finite Subset of (QuasiAdjs MaxConstrSign) holds constrs (A ast q) = (constrs q) \/ (union { (constrs a) where a is quasi-adjective : a in A } ) ;
t49_abcmiz_a:: for a being quasi-adjective for T being quasi-type holds constrs (a ast T) = (constrs a) \/ (constrs T)
t5_abcmiz_a:: for S1, S2 being standardized ConstructorSignature st the carrier' of S1 = the carrier' of S2 holds ManySortedSign(# the carrier of S1, the carrier' of S1, the Arity of S1, the ResultSort of S1 #) = ManySortedSign(# the carrier of S2, the carrier' of S2, the Arity of S2, the ResultSort of S2 #)
t50_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2, t3 being expression of C st t1 matches_with t2 & t2 matches_with t3 holds t1 matches_with t3
t51_abcmiz_a:: for C being initialized ConstructorSignature for A1, A2, A3 being Subset of (QuasiAdjs C) st A1 matches_with A2 & A2 matches_with A3 holds A1 matches_with A3
t52_abcmiz_a:: for C being initialized ConstructorSignature for T1, T2, T3 being quasi-type of C st T1 matches_with T2 & T2 matches_with T3 holds T1 matches_with T3
t53_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2 being expression of C for f being valuation of C st f unifies t1,t2 holds f unifies t2,t1
t54_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2 being expression of C st t1,t2 are_unifiable holds t1,t2 are_weakly-unifiable
t55_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2, t being expression of C st t is_a_unification_of t1,t2 holds t is_a_unification_of t2,t1
t56_abcmiz_a:: for C being initialized ConstructorSignature for t1, t2, t being expression of C st t is_a_unification_of t1,t2 holds ( t matches_with t1 & t matches_with t2 )
t57_abcmiz_a:: for n being Nat for s being SortSymbol of MaxConstrSign ex m being constructor OperSymbol of s st len (the_arity_of m) = n
t58_abcmiz_a:: for l being quasi-loci for s being SortSymbol of MaxConstrSign for m being constructor OperSymbol of s st len (the_arity_of m) = len l holds variables_in (m -trm (args l)) = rng l
t59_abcmiz_a:: for X being finite Subset of Vars st varcl X = X holds for s being SortSymbol of MaxConstrSign ex m being constructor OperSymbol of s ex p being FinSequence of QuasiTerms MaxConstrSign st ( len p = len (the_arity_of m) & vars (m -trm p) = X )
t6_abcmiz_a:: for C being ConstructorSignature holds ( C is standardized iff C is Subsignature of MaxConstrSign )
t60_abcmiz_a:: for l being empty quasi-loci holds {} is type-distribution of l
t7_abcmiz_a:: for x being Element of Vars st vars x is natural holds vars x = 0
t8_abcmiz_a:: Vars misses Constructors
t9_abcmiz_a:: for x being Element of Vars holds ( x <> * & x <> non_op ) ;
d1_abian:: for i being Integer holds ( i is even iff 2 divides i );
d2_abian:: for n being Element of NAT holds ( n is even iff ex k being Element of NAT st n = 2 * k );
d3_abian:: for x being set for f being Function holds ( x is_a_fixpoint_of f iff ( x in dom f & x = f . x ) );
d4_abian:: for A being non empty set for a being Element of A for f being Function of A,A holds ( a is_a_fixpoint_of f iff a = f . a );
d5_abian:: for f being Function holds ( f is with_fixpoint iff ex x being set st x is_a_fixpoint_of f );
d6_abian:: for X being set for x being Element of X holds ( x is covering iff union x = union (union X) );
d7_abian:: for E being set for f being Function of E,E for b3 being Equivalence_Relation of E holds ( b3 = =_ f iff for x, y being set st x in E & y in E holds ( [x,y] in b3 iff ex k, l being Element of NAT st (iter (f,k)) . x = (iter (f,l)) . y ) );
t1_abian:: for i being Integer holds ( i is odd iff ex j being Integer st i = (2 * j) + 1 )
t10_abian:: for n being Element of NAT for A being non empty set for f being Function of A,A for x being Element of A holds (iter (f,(n + 1))) . x = f . ((iter (f,n)) . x)
t11_abian:: for i being Integer holds ( i is even iff ex j being Integer st i = 2 * j ) by Lm1;
t12_abian:: for n being odd Nat holds 1 <= n
t2_abian:: for S being non empty Subset of NAT st 0 in S holds min S = 0
t3_abian:: for E being non empty set for f being Function of E,E for x being Element of E holds (iter (f,0)) . x = x
t4_abian:: for E being set for sE being Subset-Family of E holds ( sE is covering iff union sE = E )
t5_abian:: for E being set for f being Function of E,E for sE being non empty covering Subset-Family of E st ( for X being Element of sE holds X misses f .: X ) holds f is without_fixpoints
t6_abian:: for E being non empty set for f being Function of E,E for c being Element of Class (=_ f) for e being Element of c holds f . e in c
t7_abian:: for E being non empty set for f being Function of E,E for c being Element of Class (=_ f) for e being Element of c for n being Element of NAT holds (iter (f,n)) . e in c
t8_abian:: for E being non empty set for f being Function of E,E st f is without_fixpoints holds ex E1, E2, E3 being set st ( (E1 \/ E2) \/ E3 = E & f .: E1 misses E1 & f .: E2 misses E2 & f .: E3 misses E3 )
t9_abian:: for n being Nat holds ( n is odd iff ex k being Element of NAT st n = (2 * k) + 1 )
d1_absvalue:: for x being real number holds ( ( 0 <= x implies |.x.| = x ) & ( not 0 <= x implies |.x.| = - x ) );
d2_absvalue:: for x being real number holds ( ( 0 < x implies sgn x = 1 ) & ( x < 0 implies sgn x = - 1 ) & ( not 0 < x & not x < 0 implies sgn x = 0 ) );
t1_absvalue:: for x being real number holds ( abs x = x or abs x = - x ) by Def1;
t10_absvalue:: for x, y being real number st 0 < x / y holds sqrt (x / y) = (sqrt (abs x)) / (sqrt (abs y))
t11_absvalue:: for x, y being real number st 0 <= x * y holds abs (x + y) = (abs x) + (abs y)
t12_absvalue:: for x, y being real number st abs (x + y) = (abs x) + (abs y) holds 0 <= x * y
t13_absvalue:: for x, y being real number holds (abs (x + y)) / (1 + (abs (x + y))) <= ((abs x) / (1 + (abs x))) + ((abs y) / (1 + (abs y)))
t14_absvalue:: for x being real number st sgn x = 1 holds 0 < x
t15_absvalue:: for x being real number st sgn x = - 1 holds x < 0
t16_absvalue:: for x being real number st sgn x = 0 holds x = 0
t17_absvalue:: for x being real number holds x = (abs x) * (sgn x)
t18_absvalue:: for x, y being real number holds sgn (x * y) = (sgn x) * (sgn y)
t19_absvalue:: canceled;
t2_absvalue:: for x being real number holds ( x = 0 iff abs x = 0 ) by Def1, COMPLEX1:47;
t20_absvalue:: for x, y being real number holds sgn (x + y) <= ((sgn x) + (sgn y)) + 1
t21_absvalue:: for x being real number st x <> 0 holds (sgn x) * (sgn (1 / x)) = 1
t22_absvalue:: for x being real number holds 1 / (sgn x) = sgn (1 / x)
t23_absvalue:: for x, y being real number holds ((sgn x) + (sgn y)) - 1 <= sgn (x + y)
t24_absvalue:: for x being real number holds sgn x = sgn (1 / x)
t25_absvalue:: for x, y being real number holds sgn (x / y) = (sgn x) / (sgn y)
t26_absvalue:: for r being real number holds 0 <= r + (abs r)
t27_absvalue:: for r being real number holds 0 <= (- r) + (abs r)
t28_absvalue:: for r, s being real number holds ( not abs r = abs s or r = s or r = - s )
t29_absvalue:: for m being Nat holds m = abs m
t3_absvalue:: for x being real number st abs x = - x & x <> 0 holds x < 0 by Def1;
t30_absvalue:: for r being real number st r <= 0 holds abs r = - r
t4_absvalue:: for x being real number holds ( - (abs x) <= x & x <= abs x )
t5_absvalue:: for y, x being real number holds ( ( - y <= x & x <= y ) iff abs x <= y )
t6_absvalue:: for x being real number st x <> 0 holds (abs x) * (abs (1 / x)) = 1
t7_absvalue:: for x being real number holds abs (1 / x) = 1 / (abs x) by COMPLEX1:80;
t8_absvalue:: for x, y being real number st 0 <= x * y holds sqrt (x * y) = (sqrt (abs x)) * (sqrt (abs y))
t9_absvalue:: for x, z, y, t being real number st abs x <= z & abs y <= t holds abs (x + y) <= z + t
d1_aff_1:: for AS being AffinSpace for a, b, c being Element of AS holds ( LIN a,b,c iff a,b // a,c );
d2_aff_1:: for AS being AffinSpace for a, b being Element of AS for b4 being Subset of AS holds ( b4 = Line (a,b) iff for x being Element of AS holds ( x in b4 iff LIN a,b,x ) );
d3_aff_1:: for AS being AffinSpace for A being Subset of AS holds ( A is being_line iff ex a, b being Element of AS st ( a <> b & A = Line (a,b) ) );
d4_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS holds ( a,b // A iff ex c, d being Element of AS st ( c <> d & A = Line (c,d) & a,b // c,d ) );
d5_aff_1:: for AS being AffinSpace for A, C being Subset of AS holds ( A // C iff ex a, b being Element of AS st ( A = Line (a,b) & a <> b & a,b // C ) );
t1_aff_1:: for AS being AffinSpace for a being Element of AS ex b being Element of AS st a <> b
t10_aff_1:: for AS being AffinSpace for x, y, z, t being Element of AS st LIN x,y,z & LIN x,y,t holds x,y // z,t
t11_aff_1:: for AS being AffinSpace for u, z, x, y, w being Element of AS st u <> z & LIN x,y,u & LIN x,y,z & LIN u,z,w holds LIN x,y,w
t12_aff_1:: for AS being AffinSpace holds not for x, y, z being Element of AS holds LIN x,y,z
t13_aff_1:: for AS being AffinSpace for x, y being Element of AS st x <> y holds ex z being Element of AS st not LIN x,y,z
t14_aff_1:: for AS being AffinSpace for o, a, b, b9 being Element of AS st not LIN o,a,b & LIN o,b,b9 & a,b // a,b9 holds b = b9
t15_aff_1:: for AS being AffinSpace for a, b being Element of AS holds ( a in Line (a,b) & b in Line (a,b) )
t16_aff_1:: for AS being AffinSpace for c, a, b, d being Element of AS st c in Line (a,b) & d in Line (a,b) & c <> d holds Line (c,d) c= Line (a,b)
t17_aff_1:: for AS being AffinSpace for c, a, b, d being Element of AS st c in Line (a,b) & d in Line (a,b) & a <> b holds Line (a,b) c= Line (c,d)
t18_aff_1:: for AS being AffinSpace for a, b being Element of AS for A, C being Subset of AS st A is being_line & C is being_line & a in A & b in A & a in C & b in C & not a = b holds A = C
t19_aff_1:: for AS being AffinSpace for A being Subset of AS st A is being_line holds ex a, b being Element of AS st ( a in A & b in A & a <> b )
t2_aff_1:: for AS being AffinSpace for x, y being Element of AS holds ( x,y // y,x & x,y // x,y )
t20_aff_1:: for AS being AffinSpace for a being Element of AS for A being Subset of AS st A is being_line holds ex b being Element of AS st ( a <> b & b in A )
t21_aff_1:: for AS being AffinSpace for a, b, c being Element of AS holds ( LIN a,b,c iff ex A being Subset of AS st ( A is being_line & a in A & b in A & c in A ) )
t22_aff_1:: for AS being AffinSpace for c, a, b, d being Element of AS st c in Line (a,b) & a <> b holds ( d in Line (a,b) iff a,b // c,d )
t23_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS st A is being_line & a in A holds ( b in A iff a,b // A )
t24_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS holds ( a <> b & A = Line (a,b) iff ( A is being_line & a in A & b in A & a <> b ) ) by Def3, Lm6, Th15;
t25_aff_1:: for AS being AffinSpace for a, b, x being Element of AS for A being Subset of AS st A is being_line & a in A & b in A & a <> b & LIN a,b,x holds x in A
t26_aff_1:: for AS being AffinSpace for A being Subset of AS st ex a, b being Element of AS st a,b // A holds A is being_line
t27_aff_1:: for AS being AffinSpace for c, d, a, b being Element of AS for A being Subset of AS st c in A & d in A & A is being_line & c <> d holds ( a,b // A iff a,b // c,d )
t28_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS st a,b // A holds ex c, d being Element of AS st ( c <> d & c in A & d in A & a,b // c,d )
t29_aff_1:: for AS being AffinSpace for a, b being Element of AS st a <> b holds a,b // Line (a,b)
t3_aff_1:: for AS being AffinSpace for x, y, z being Element of AS holds ( x,y // z,z & z,z // x,y )
t30_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being being_line Subset of AS holds ( a,b // A iff ex c, d being Element of AS st ( c <> d & c in A & d in A & a,b // c,d ) )
t31_aff_1:: for AS being AffinSpace for a, b, c, d being Element of AS for A being being_line Subset of AS st a,b // A & c,d // A holds a,b // c,d
t32_aff_1:: for AS being AffinSpace for a, b, p, q being Element of AS for A being Subset of AS st a,b // A & a,b // p,q & a <> b holds p,q // A
t33_aff_1:: for AS being AffinSpace for a being Element of AS for A being being_line Subset of AS holds a,a // A
t34_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS st a,b // A holds b,a // A
t35_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS st a,b // A & not a in A holds not b in A
t36_aff_1:: for AS being AffinSpace for A, C being Subset of AS st A // C holds ( A is being_line & C is being_line )
t37_aff_1:: for AS being AffinSpace for A, C being Subset of AS holds ( A // C iff ex a, b, c, d being Element of AS st ( a <> b & c <> d & a,b // c,d & A = Line (a,b) & C = Line (c,d) ) )
t38_aff_1:: for AS being AffinSpace for a, b, c, d being Element of AS for A, C being being_line Subset of AS st a in A & b in A & c in C & d in C & a <> b & c <> d holds ( A // C iff a,b // c,d )
t39_aff_1:: for AS being AffinSpace for a, b, c, d being Element of AS for A, C being Subset of AS st a in A & b in A & c in C & d in C & A // C holds a,b // c,d
t4_aff_1:: for AS being AffinSpace for x, y, z, t being Element of AS st x,y // z,t holds ( x,y // t,z & y,x // z,t & y,x // t,z & z,t // x,y & z,t // y,x & t,z // x,y & t,z // y,x )
t40_aff_1:: for AS being AffinSpace for a, b being Element of AS for A, C being Subset of AS st a in A & b in A & A // C holds a,b // C
t41_aff_1:: for AS being AffinSpace for A being being_line Subset of AS holds A // A
t42_aff_1:: for AS being AffinSpace for A, C being Subset of AS st A // C holds C // A
t43_aff_1:: for AS being AffinSpace for a, b being Element of AS for A, C being Subset of AS st a,b // A & A // C holds a,b // C
t44_aff_1:: for AS being AffinSpace for A, C, D being Subset of AS st ( ( A // C & C // D ) or ( A // C & D // C ) or ( C // A & C // D ) or ( C // A & D // C ) ) holds A // D by Lm7;
t45_aff_1:: for AS being AffinSpace for p being Element of AS for A, C being Subset of AS st A // C & p in A & p in C holds A = C
t46_aff_1:: for AS being AffinSpace for x, a, b being Element of AS for K being Subset of AS st x in K & not a in K & a,b // K & not a = b holds not LIN x,a,b
t47_aff_1:: for AS being AffinSpace for a9, b9, p, a, b being Element of AS for K being Subset of AS st a9,b9 // K & LIN p,a,a9 & LIN p,b,b9 & p in K & not a in K & a = b holds a9 = b9
t48_aff_1:: for AS being AffinSpace for a, b, c, d being Element of AS for A being being_line Subset of AS st a in A & b in A & c in A & a <> b & a,b // c,d holds d in A
t49_aff_1:: for AS being AffinSpace for a being Element of AS for A being being_line Subset of AS ex C being Subset of AS st ( a in C & A // C )
t5_aff_1:: for AS being AffinSpace for a, b, x, y, z, t being Element of AS st a <> b & ( ( a,b // x,y & a,b // z,t ) or ( a,b // x,y & z,t // a,b ) or ( x,y // a,b & z,t // a,b ) or ( x,y // a,b & a,b // z,t ) ) holds x,y // z,t
t50_aff_1:: for AS being AffinSpace for p being Element of AS for A, C, D being Subset of AS st A // C & A // D & p in C & p in D holds C = D by Lm7, Th45;
t51_aff_1:: for AS being AffinSpace for a, b, c, d being Element of AS for A being Subset of AS st A is being_line & a in A & b in A & c in A & d in A holds a,b // c,d by Th39, Th41;
t52_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS st A is being_line & a in A & b in A holds a,b // A by Th23;
t53_aff_1:: for AS being AffinSpace for a, b being Element of AS for A, C being Subset of AS st a,b // A & a,b // C & a <> b holds A // C
t54_aff_1:: for AS being AffinSpace for o, a, b, a9, b9 being Element of AS st not LIN o,a,b & LIN o,a,a9 & LIN o,b,b9 & a9 = b9 holds ( a9 = o & b9 = o )
t55_aff_1:: for AS being AffinSpace for o, a, b, b9, a9 being Element of AS st not LIN o,a,b & LIN o,b,b9 & a,b // a9,b9 & a9 = o holds b9 = o
t56_aff_1:: for AS being AffinSpace for o, a, b, a9, b9, x being Element of AS st not LIN o,a,b & LIN o,a,a9 & LIN o,b,b9 & LIN o,b,x & a,b // a9,b9 & a,b // a9,x holds b9 = x
t57_aff_1:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS st A is being_line & a in A & b in A & a <> b holds A = Line (a,b) by Lm6;
t58_aff_1:: for AP being AffinPlane for A, C being Subset of AP st A is being_line & C is being_line & not A // C holds ex x being Element of AP st ( x in A & x in C )
t59_aff_1:: for AP being AffinPlane for a, b being Element of AP for A being Subset of AP st A is being_line & not a,b // A holds ex x being Element of AP st ( x in A & LIN a,b,x )
t6_aff_1:: for AS being AffinSpace for x, y, z being Element of AS st LIN x,y,z holds ( LIN x,z,y & LIN y,x,z & LIN y,z,x & LIN z,x,y & LIN z,y,x )
t60_aff_1:: for AP being AffinPlane for a, b, c, d being Element of AP st not a,b // c,d holds ex p being Element of AP st ( LIN a,b,p & LIN c,d,p )
t7_aff_1:: for AS being AffinSpace for x, y being Element of AS holds ( LIN x,x,y & LIN x,y,y & LIN x,y,x )
t8_aff_1:: for AS being AffinSpace for x, y, z, t, u being Element of AS st x <> y & LIN x,y,z & LIN x,y,t & LIN x,y,u holds LIN z,t,u
t9_aff_1:: for AS being AffinSpace for x, y, z, t being Element of AS st x <> y & LIN x,y,z & x,y // z,t holds LIN x,y,t
d1_aff_2:: for AP being AffinPlane holds ( AP is satisfying_PPAP iff for M, N being Subset of AP for a, b, c, a9, b9, c9 being Element of AP st M is being_line & N is being_line & a in M & b in M & c in M & a9 in N & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 holds a,c9 // c,a9 );
d10_aff_2:: for AP being AffinPlane holds ( AP is satisfying_TDES_3 iff for K being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st K is being_line & o in K & c in K & not a in K & o <> c & a <> b & LIN o,a,a9 & LIN o,b,b9 & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & a,b // K holds c9 in K );
d11_aff_2:: for AP being AffinSpace holds ( AP is translational iff for A, P, C being Subset of AP for a, b, c, a9, b9, c9 being Element of AP st A // P & A // C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9 );
d12_aff_2:: for AP being AffinPlane holds ( AP is satisfying_des_1 iff for A, P, C being Subset of AP for a, b, c, a9, b9, c9 being Element of AP st A // P & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & not LIN a,b,c & c <> c9 holds A // C );
d13_aff_2:: for AP being AffinSpace holds ( AP is satisfying_pap iff for M, N being Subset of AP for a, b, c, a9, b9, c9 being Element of AP st M is being_line & N is being_line & a in M & b in M & c in M & M // N & M <> N & a9 in N & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 holds a,c9 // c,a9 );
d14_aff_2:: for AP being AffinPlane holds ( AP is satisfying_pap_1 iff for M, N being Subset of AP for a, b, c, a9, b9, c9 being Element of AP st M is being_line & N is being_line & a in M & b in M & c in M & M // N & M <> N & a9 in N & b9 in N & a,b9 // b,a9 & b,c9 // c,b9 & a,c9 // c,a9 & a9 <> b9 holds c9 in N );
d2_aff_2:: for AP being AffinSpace holds ( AP is Pappian iff for M, N being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st M is being_line & N is being_line & M <> N & o in M & o in N & o <> a & o <> a9 & o <> b & o <> b9 & o <> c & o <> c9 & a in M & b in M & c in M & a9 in N & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 holds a,c9 // c,a9 );
d3_aff_2:: for AP being AffinPlane holds ( AP is satisfying_PAP_1 iff for M, N being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st M is being_line & N is being_line & M <> N & o in M & o in N & o <> a & o <> a9 & o <> b & o <> b9 & o <> c & o <> c9 & a in M & b in M & c in M & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 & a,c9 // c,a9 & b <> c holds a9 in N );
d4_aff_2:: for AP being AffinSpace holds ( AP is Desarguesian iff for A, P, C being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9 );
d5_aff_2:: for AP being AffinPlane holds ( AP is satisfying_DES_1 iff for A, P, C being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st o in A & o in P & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 & not LIN a,b,c & c <> c9 holds o in C );
d6_aff_2:: for AP being AffinPlane holds ( AP is satisfying_DES_2 iff for A, P, C being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & a9 in A & b in P & b9 in P & c in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & b,c // b9,c9 holds c9 in C );
d7_aff_2:: for AP being AffinSpace holds ( AP is Moufangian iff for K being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st K is being_line & o in K & c in K & c9 in K & not a in K & o <> c & a <> b & LIN o,a,a9 & LIN o,b,b9 & a,b // a9,b9 & a,c // a9,c9 & a,b // K holds b,c // b9,c9 );
d8_aff_2:: for AP being AffinPlane holds ( AP is satisfying_TDES_1 iff for K being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st K is being_line & o in K & c in K & c9 in K & not a in K & o <> c & a <> b & LIN o,a,a9 & a,b // a9,b9 & b,c // b9,c9 & a,c // a9,c9 & a,b // K holds LIN o,b,b9 );
d9_aff_2:: for AP being AffinPlane holds ( AP is satisfying_TDES_2 iff for K being Subset of AP for o, a, b, c, a9, b9, c9 being Element of AP st K is being_line & o in K & c in K & c9 in K & not a in K & o <> c & a <> b & LIN o,a,a9 & LIN o,b,b9 & b,c // b9,c9 & a,c // a9,c9 & a,b // K holds a,b // a9,b9 );
t1_aff_2:: for AP being AffinPlane holds ( AP is Pappian iff AP is satisfying_PAP_1 )
t10_aff_2:: for AP being AffinPlane holds ( AP is satisfying_PPAP iff ( AP is Pappian & AP is satisfying_pap ) )
t11_aff_2:: for AP being AffinPlane st AP is Pappian holds AP is Desarguesian
t12_aff_2:: for AP being AffinPlane st AP is Desarguesian holds AP is Moufangian
t13_aff_2:: for AP being AffinPlane st AP is satisfying_TDES_1 holds AP is satisfying_des_1
t14_aff_2:: for AP being AffinPlane st AP is Moufangian holds AP is translational
t15_aff_2:: for AP being AffinPlane st AP is translational holds AP is satisfying_pap
t2_aff_2:: for AP being AffinPlane holds ( AP is Desarguesian iff AP is satisfying_DES_1 )
t3_aff_2:: for AP being AffinPlane st AP is Moufangian holds AP is satisfying_TDES_1
t4_aff_2:: for AP being AffinPlane st AP is satisfying_TDES_1 holds AP is satisfying_TDES_2
t5_aff_2:: for AP being AffinPlane st AP is satisfying_TDES_2 holds AP is satisfying_TDES_3
t6_aff_2:: for AP being AffinPlane st AP is satisfying_TDES_3 holds AP is Moufangian
t7_aff_2:: for AP being AffinPlane holds ( AP is translational iff AP is satisfying_des_1 )
t8_aff_2:: for AP being AffinPlane holds ( AP is satisfying_pap iff AP is satisfying_pap_1 )
t9_aff_2:: for AP being AffinPlane st AP is Pappian holds AP is satisfying_pap
d1_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES1 iff for A, P, C being Subset of AP for o, a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & P <> A & P <> C & A <> C & o in A & a in A & a9 in A & o in P & b in P & b9 in P & o in C & c in C & c9 in C & o <> a & o <> b & o <> c & p <> q & not LIN b,a,c & not LIN b9,a9,c9 & a <> a9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 holds a,c // p,q );
d2_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES1_1 iff for A, P, C being Subset of AP for o, a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & P <> A & P <> C & A <> C & o in A & a in A & a9 in A & o in P & b in P & b9 in P & o in C & c in C & c9 in C & o <> a & o <> b & o <> c & p <> q & c <> q & not LIN b,a,c & not LIN b9,a9,c9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // p,q holds a,c // a9,c9 );
d3_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES1_2 iff for A, P, C being Subset of AP for o, a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & P <> A & P <> C & A <> C & o in A & a in A & a9 in A & o in P & b in P & b9 in P & c in C & c9 in C & o <> a & o <> b & o <> c & p <> q & not LIN b,a,c & not LIN b9,a9,c9 & c <> c9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 & a,c // p,q holds o in C );
d4_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES1_3 iff for A, P, C being Subset of AP for o, a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & P <> A & P <> C & A <> C & o in A & a in A & a9 in A & b in P & b9 in P & o in C & c in C & c9 in C & o <> a & o <> b & o <> c & p <> q & not LIN b,a,c & not LIN b9,a9,c9 & b <> b9 & a <> a9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 & a,c // p,q holds o in P );
d5_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES2 iff for A, P, C being Subset of AP for a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & A <> P & A <> C & P <> C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A // P & A // C & not LIN b,a,c & not LIN b9,a9,c9 & p <> q & a <> a9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 holds a,c // p,q );
d6_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES2_1 iff for A, P, C being Subset of AP for a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & A <> P & A <> C & P <> C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A // P & A // C & not LIN b,a,c & not LIN b9,a9,c9 & p <> q & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // p,q holds a,c // a9,c9 );
d7_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES2_2 iff for A, P, C being Subset of AP for a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & A <> P & A <> C & P <> C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A // C & not LIN b,a,c & not LIN b9,a9,c9 & p <> q & a <> a9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 & a,c // p,q holds A // P );
d8_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES2_3 iff for A, P, C being Subset of AP for a, a9, b, b9, c, c9, p, q being Element of AP st A is being_line & P is being_line & C is being_line & A <> P & A <> C & P <> C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A // P & not LIN b,a,c & not LIN b9,a9,c9 & p <> q & c <> c9 & LIN b,a,p & LIN b9,a9,p & LIN b,c,q & LIN b9,c9,q & a,c // a9,c9 & a,c // p,q holds A // C );
t1_aff_3:: for AP being AffinPlane st AP is satisfying_DES1 holds AP is satisfying_DES1_1
t10_aff_3:: for AP being AffinPlane st AP is satisfying_DES1_3 holds AP is satisfying_DES2_1
t2_aff_3:: for AP being AffinPlane st AP is satisfying_DES1_1 holds AP is satisfying_DES1
t3_aff_3:: for AP being AffinPlane st AP is Desarguesian holds AP is satisfying_DES1
t4_aff_3:: for AP being AffinPlane st AP is Desarguesian holds AP is satisfying_DES1_2
t5_aff_3:: for AP being AffinPlane st AP is satisfying_DES1_2 holds AP is satisfying_DES1_3
t6_aff_3:: for AP being AffinPlane st AP is satisfying_DES1_2 holds AP is Desarguesian
t7_aff_3:: for AP being AffinPlane st AP is satisfying_DES2_1 holds AP is satisfying_DES2
t8_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES2_1 iff AP is satisfying_DES2_3 )
t9_aff_3:: for AP being AffinPlane holds ( AP is satisfying_DES2 iff AP is satisfying_DES2_2 )
d1_aff_4:: for AS being AffinSpace for K, P being Subset of AS holds Plane (K,P) = { a where a is Element of AS : ex b being Element of AS st ( a,b // K & b in P ) } ;
d2_aff_4:: for AS being AffinSpace for X being Subset of AS holds ( X is being_plane iff ex K, P being Subset of AS st ( K is being_line & P is being_line & not K // P & X = Plane (K,P) ) );
d3_aff_4:: for AS being AffinSpace for a being Element of AS for K being Subset of AS st K is being_line holds for b4 being Subset of AS holds ( b4 = a * K iff ( a in b4 & K // b4 ) );
d4_aff_4:: for AS being AffinSpace for X, Y being Subset of AS holds ( X '||' Y iff for a being Element of AS for A being Subset of AS st a in Y & A is being_line & A c= X holds a * A c= Y );
d5_aff_4:: for AS being AffinSpace for K, M, N being Subset of AS holds ( K,M,N is_coplanar iff ex X being Subset of AS st ( K c= X & M c= X & N c= X & X is being_plane ) );
d6_aff_4:: for AS being AffinSpace for a being Element of AS for X being Subset of AS st X is being_plane holds for b4 being Subset of AS holds ( b4 = a + X iff ( a in b4 & X '||' b4 & b4 is being_plane ) );
t1_aff_4:: for AS being AffinSpace for p, a, a9, b being Element of AS st ( LIN p,a,a9 or LIN p,a9,a ) & p <> a holds ex b9 being Element of AS st ( LIN p,b,b9 & a,b // a9,b9 )
t10_aff_4:: for AS being AffinSpace for a, b, b9, a9 being Element of AS for M, N being Subset of AS st ( M // N or N // M ) & a in M & b in N & b9 in N & M <> N & ( a,b // a9,b9 or b,a // b9,a9 ) & a = a9 holds b = b9
t11_aff_4:: for AS being AffinSpace for a, b being Element of AS ex A being Subset of AS st ( a in A & b in A & A is being_line )
t12_aff_4:: for AS being AffinSpace for A being Subset of AS st A is being_line holds ex q being Element of AS st not q in A
t13_aff_4:: for AS being AffinSpace for K, P being Subset of AS st not K is being_line holds Plane (K,P) = {}
t14_aff_4:: for AS being AffinSpace for K, P being Subset of AS st K is being_line holds P c= Plane (K,P)
t15_aff_4:: for AS being AffinSpace for K, P being Subset of AS st K // P holds Plane (K,P) = P
t16_aff_4:: for AS being AffinSpace for K, M, P being Subset of AS st K // M holds Plane (K,P) = Plane (M,P)
t17_aff_4:: for AS being AffinSpace for p, a, b, a9, b9 being Element of AS for M, N, P, Q being Subset of AS st p in M & a in M & b in M & p in N & a9 in N & b9 in N & not p in P & not p in Q & M <> N & a in P & a9 in P & b in Q & b9 in Q & M is being_line & N is being_line & P is being_line & Q is being_line & not P // Q holds ex q being Element of AS st ( q in P & q in Q )
t18_aff_4:: for AS being AffinSpace for a, b, a9, b9 being Element of AS for M, N, P, Q being Subset of AS st a in M & b in M & a9 in N & b9 in N & a in P & a9 in P & b in Q & b9 in Q & M <> N & M // N & P is being_line & Q is being_line & not P // Q holds ex q being Element of AS st ( q in P & q in Q )
t19_aff_4:: for AS being AffinSpace for a, b being Element of AS for X being Subset of AS st X is being_plane & a in X & b in X & a <> b holds Line (a,b) c= X
t2_aff_4:: for AS being AffinSpace for a, b being Element of AS for A being Subset of AS st ( a,b // A or b,a // A ) & a in A holds b in A
t20_aff_4:: for AS being AffinSpace for K, P, Q being Subset of AS st K is being_line & P is being_line & Q is being_line & not K // Q & Q c= Plane (K,P) holds Plane (K,Q) = Plane (K,P)
t21_aff_4:: for AS being AffinSpace for K, P, Q being Subset of AS st K is being_line & P is being_line & Q is being_line & Q c= Plane (K,P) & not P // Q holds ex q being Element of AS st ( q in P & q in Q )
t22_aff_4:: for AS being AffinSpace for X, M, N being Subset of AS st X is being_plane & M is being_line & N is being_line & M c= X & N c= X & not M // N holds ex q being Element of AS st ( q in M & q in N )
t23_aff_4:: for AS being AffinSpace for a being Element of AS for X, M, N being Subset of AS st X is being_plane & a in X & M c= X & a in N & ( M // N or N // M ) holds N c= X
t24_aff_4:: for AS being AffinSpace for a, b being Element of AS for X, Y being Subset of AS st X is being_plane & Y is being_plane & a in X & b in X & a in Y & b in Y & X <> Y & a <> b holds X /\ Y is being_line
t25_aff_4:: for AS being AffinSpace for a, b, c being Element of AS for X, Y being Subset of AS st X is being_plane & Y is being_plane & a in X & b in X & c in X & a in Y & b in Y & c in Y & not LIN a,b,c holds X = Y
t26_aff_4:: for AS being AffinSpace for X, Y, M, N being Subset of AS st X is being_plane & Y is being_plane & M is being_line & N is being_line & M c= X & N c= X & M c= Y & N c= Y & M <> N holds X = Y
t27_aff_4:: for AS being AffinSpace for a being Element of AS for A being Subset of AS st A is being_line holds a * A is being_line
t28_aff_4:: for AS being AffinSpace for a being Element of AS for X, M being Subset of AS st X is being_plane & M is being_line & a in X & M c= X holds a * M c= X
t29_aff_4:: for AS being AffinSpace for a, b, c, d being Element of AS for X being Subset of AS st X is being_plane & a in X & b in X & c in X & a,b // c,d & a <> b holds d in X
t3_aff_4:: for AS being AffinSpace for a, b being Element of AS for A, K being Subset of AS st ( a,b // A or b,a // A ) & A // K holds ( a,b // K & b,a // K )
t30_aff_4:: for AS being AffinSpace for a being Element of AS for A being Subset of AS st A is being_line holds ( a in A iff a * A = A )
t31_aff_4:: for AS being AffinSpace for a, q being Element of AS for A being Subset of AS st A is being_line holds a * A = a * (q * A)
t32_aff_4:: for AS being AffinSpace for a being Element of AS for K, M being Subset of AS st K // M holds a * K = a * M
t33_aff_4:: for AS being AffinSpace for X, Y being Subset of AS st X c= Y & ( ( X is being_line & Y is being_line ) or ( X is being_plane & Y is being_plane ) ) holds X = Y
t34_aff_4:: for AS being AffinSpace for X being Subset of AS st X is being_plane holds ex a, b, c being Element of AS st ( a in X & b in X & c in X & not LIN a,b,c )
t35_aff_4:: for AS being AffinSpace for M, X being Subset of AS st M is being_line & X is being_plane holds ex q being Element of AS st ( q in X & not q in M )
t36_aff_4:: for AS being AffinSpace for a being Element of AS for A being Subset of AS st A is being_line holds ex X being Subset of AS st ( a in X & A c= X & X is being_plane )
t37_aff_4:: for AS being AffinSpace for a, b, c being Element of AS ex X being Subset of AS st ( a in X & b in X & c in X & X is being_plane )
t38_aff_4:: for AS being AffinSpace for q being Element of AS for M, N being Subset of AS st q in M & q in N & M is being_line & N is being_line holds ex X being Subset of AS st ( M c= X & N c= X & X is being_plane )
t39_aff_4:: for AS being AffinSpace for M, N being Subset of AS st M // N holds ex X being Subset of AS st ( M c= X & N c= X & X is being_plane )
t4_aff_4:: for AS being AffinSpace for a, b, c, d being Element of AS for A being Subset of AS st ( a,b // A or b,a // A ) & ( a,b // c,d or c,d // a,b ) & a <> b holds ( c,d // A & d,c // A )
t40_aff_4:: for AS being AffinSpace for M, N being Subset of AS st M is being_line & N is being_line holds ( M // N iff M '||' N )
t41_aff_4:: for AS being AffinSpace for M, X being Subset of AS st M is being_line & X is being_plane holds ( M '||' X iff ex N being Subset of AS st ( N c= X & ( M // N or N // M ) ) )
t42_aff_4:: for AS being AffinSpace for M, X being Subset of AS st M is being_line & X is being_plane & M c= X holds M '||' X
t43_aff_4:: for AS being AffinSpace for a being Element of AS for A, X being Subset of AS st A is being_line & X is being_plane & a in A & a in X & A '||' X holds A c= X
t44_aff_4:: for AS being AffinSpace for K, M, N being Subset of AS st K,M,N is_coplanar holds ( K,N,M is_coplanar & M,K,N is_coplanar & M,N,K is_coplanar & N,K,M is_coplanar & N,M,K is_coplanar )
t45_aff_4:: for AS being AffinSpace for M, N, K, A being Subset of AS st M is being_line & N is being_line & M,N,K is_coplanar & M,N,A is_coplanar & M <> N holds M,K,A is_coplanar
t46_aff_4:: for AS being AffinSpace for K, M, X, A being Subset of AS st K is being_line & M is being_line & X is being_plane & K c= X & M c= X & K <> M holds ( K,M,A is_coplanar iff A c= X )
t47_aff_4:: for AS being AffinSpace for q being Element of AS for K, M being Subset of AS st q in K & q in M & K is being_line & M is being_line holds ( K,M,M is_coplanar & M,K,M is_coplanar & M,M,K is_coplanar )
t48_aff_4:: for AS being AffinSpace for X being Subset of AS st AS is not AffinPlane & X is being_plane holds ex q being Element of AS st not q in X
t49_aff_4:: for AS being AffinSpace for q, a, b, c, a9, b9, c9 being Element of AS for A, P, C being Subset of AS st AS is not AffinPlane & q in A & q in P & q in C & q <> a & q <> b & q <> c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9
t5_aff_4:: for AS being AffinSpace for a, b being Element of AS for M, N being Subset of AS st ( a,b // M or b,a // M ) & ( a,b // N or b,a // N ) & a <> b holds M // N
t50_aff_4:: for AS being AffinSpace st AS is not AffinPlane holds AS is Desarguesian
t51_aff_4:: for AS being AffinSpace for a, a9, b, b9, c, c9 being Element of AS for A, P, C being Subset of AS st AS is not AffinPlane & A // P & A // C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9
t52_aff_4:: for AS being AffinSpace st AS is not AffinPlane holds AS is translational
t53_aff_4:: for AS being AffinSpace for a, b, c, a9, b9 being Element of AS st AS is AffinPlane & not LIN a,b,c holds ex c9 being Element of AS st ( a,c // a9,c9 & b,c // b9,c9 )
t54_aff_4:: for AS being AffinSpace for a, b, c, a9, b9 being Element of AS st not LIN a,b,c & a9 <> b9 & a,b // a9,b9 holds ex c9 being Element of AS st ( a,c // a9,c9 & b,c // b9,c9 )
t55_aff_4:: for AS being AffinSpace for X, Y being Subset of AS st X is being_plane & Y is being_plane holds ( X '||' Y iff ex A, P, M, N being Subset of AS st ( not A // P & A c= X & P c= X & M c= Y & N c= Y & ( A // M or M // A ) & ( P // N or N // P ) ) )
t56_aff_4:: for AS being AffinSpace for A, M, X being Subset of AS st A // M & M '||' X holds A '||' X
t57_aff_4:: for AS being AffinSpace for X being Subset of AS st X is being_plane holds X '||' X
t58_aff_4:: for AS being AffinSpace for X, Y being Subset of AS st X is being_plane & Y is being_plane & X '||' Y holds Y '||' X
t59_aff_4:: for AS being AffinSpace for X being Subset of AS st X is being_plane holds X <> {}
t6_aff_4:: for AS being AffinSpace for a, b, c, d being Element of AS for M being Subset of AS st ( a,b // M or b,a // M ) & ( c,d // M or d,c // M ) holds a,b // c,d
t60_aff_4:: for AS being AffinSpace for X, Y, Z being Subset of AS st X '||' Y & Y '||' Z & Y <> {} holds X '||' Z
t61_aff_4:: for AS being AffinSpace for X, Y, Z being Subset of AS st X is being_plane & Y is being_plane & Z is being_plane & ( ( X '||' Y & Y '||' Z ) or ( X '||' Y & Z '||' Y ) or ( Y '||' X & Y '||' Z ) or ( Y '||' X & Z '||' Y ) ) holds X '||' Z
t62_aff_4:: for AS being AffinSpace for a being Element of AS for X, Y being Subset of AS st X is being_plane & Y is being_plane & a in X & a in Y & X '||' Y holds X = Y by Lm13;
t63_aff_4:: for AS being AffinSpace for a, b, c, d being Element of AS for X, Y, Z being Subset of AS st X is being_plane & Y is being_plane & Z is being_plane & X '||' Y & X <> Y & a in X /\ Z & b in X /\ Z & c in Y /\ Z & d in Y /\ Z holds a,b // c,d
t64_aff_4:: for AS being AffinSpace for a, b, c, d being Element of AS for X, Y, Z being Subset of AS st X is being_plane & Y is being_plane & Z is being_plane & X '||' Y & a in X /\ Z & b in X /\ Z & c in Y /\ Z & d in Y /\ Z & X <> Y & a <> b & c <> d holds X /\ Z // Y /\ Z
t65_aff_4:: for AS being AffinSpace for a being Element of AS for X being Subset of AS st X is being_plane holds ex Y being Subset of AS st ( a in Y & X '||' Y & Y is being_plane )
t66_aff_4:: for AS being AffinSpace for a being Element of AS for X being Subset of AS st X is being_plane holds ( a in X iff a + X = X )
t67_aff_4:: for AS being AffinSpace for a, q being Element of AS for X being Subset of AS st X is being_plane holds a + X = a + (q + X)
t68_aff_4:: for AS being AffinSpace for a being Element of AS for A, X being Subset of AS st A is being_line & X is being_plane & A '||' X holds a * A c= a + X
t69_aff_4:: for AS being AffinSpace for a being Element of AS for X, Y being Subset of AS st X is being_plane & Y is being_plane & X '||' Y holds a + X = a + Y
t7_aff_4:: for AS being AffinSpace for a, b, c, d being Element of AS for A, C being Subset of AS st ( A // C or C // A ) & a <> b & ( a,b // c,d or c,d // a,b ) & a in A & b in A & c in C holds d in C
t8_aff_4:: for AS being AffinSpace for q, a, b, b9, a9 being Element of AS for M, N being Subset of AS st q in M & q in N & a in M & b in N & b9 in N & q <> a & q <> b & M <> N & ( a,b // a9,b9 or b,a // b9,a9 ) & M is being_line & N is being_line & q = a9 holds q = b9
t9_aff_4:: for AS being AffinSpace for q, a, a9, b, b9 being Element of AS for M, N being Subset of AS st q in M & q in N & a in M & a9 in M & b in N & b9 in N & q <> a & q <> b & M <> N & ( a,b // a9,b9 or b,a // b9,a9 ) & M is being_line & N is being_line & a = a9 holds b = b9
d1_afinsq_1:: for x being set holds <%x%> = 0 .--> x;
d10_afinsq_1:: for D being non empty set for q being FinSequence of D for n being Nat st n > len q & NAT c= D holds for b4 being non empty XFinSequence of D holds ( b4 = FS2XFS* (q,n) iff ( len q = b4 . 0 & len b4 = n & ( for i being Nat st 1 <= i & i <= len q holds b4 . i = q . i ) & ( for j being Nat st len q < j & j < n holds b4 . j = 0 ) ) );
d11_afinsq_1:: for D being non empty set for p being XFinSequence of D st p . 0 is Nat & p . 0 in len p holds for b3 being FinSequence of D holds ( b3 = XFS2FS* p iff for m being Nat st m = p . 0 holds ( len b3 = m & ( for i being Nat st 1 <= i & i <= m holds b3 . i = p . i ) ) );
d12_afinsq_1:: for F being Function holds ( F is initial iff for m, n being Nat st n in dom F & m < n holds m in dom F );
d13_afinsq_1:: for D being set for f being XFinSequence of D holds Down f = f;
d14_afinsq_1:: for x1, x2, x3, x4 being set holds <%x1,x2,x3,x4%> = ((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>;
d2_afinsq_1:: for D being set holds <%> D = {} ;
d3_afinsq_1:: for p, q being XFinSequence for b3 being set holds ( b3 = p ^ q iff ( dom b3 = (len p) + (len q) & ( for k being Nat st k in dom p holds b3 . k = p . k ) & ( for k being Nat st k in dom q holds b3 . ((len p) + k) = q . k ) ) );
d4_afinsq_1:: for x being set for b2 being Function holds ( b2 = <%x%> iff ( dom b2 = 1 & b2 . 0 = x ) );
d5_afinsq_1:: for x, y being set holds <%x,y%> = <%x%> ^ <%y%>;
d6_afinsq_1:: for x, y, z being set holds <%x,y,z%> = (<%x%> ^ <%y%>) ^ <%z%>;
d7_afinsq_1:: for D, b2 being set holds ( b2 = D ^omega iff for x being set holds ( x in b2 iff x is XFinSequence of D ) );
d8_afinsq_1:: for D being set for q being FinSequence of D for b3 being XFinSequence of D holds ( b3 = FS2XFS q iff ( len b3 = len q & ( for i being Nat st i < len q holds q . (i + 1) = b3 . i ) ) );
d9_afinsq_1:: for D being set for q being XFinSequence of D for b3 being FinSequence of D holds ( b3 = XFS2FS q iff ( len b3 = len q & ( for i being Nat st 1 <= i & i <= len q holds q . (i -' 1) = b3 . i ) ) );
s1_afinsq_1:: scheme XSeqEx{ F1() -> Nat, P1[ set , set ] } : ex p being XFinSequence st ( dom p = F1() & ( for k being Nat st k in F1() holds P1[k,p . k] ) ) provided A1: for k being Nat st k in F1() holds ex x being set st P1[k,x]
s2_afinsq_1:: scheme XSeqLambda{ F1() -> Nat, F2( set ) -> set } : ex p being XFinSequence st ( len p = F1() & ( for k being Nat st k in F1() holds p . k = F2(k) ) )
s3_afinsq_1:: scheme IndXSeq{ P1[ XFinSequence] } : for p being XFinSequence holds P1[p] provided A1: P1[ {} ] and A2: for p being XFinSequence for x being set st P1[p] holds P1[p ^ <%x%>]
s4_afinsq_1:: scheme SepXSeq{ F1() -> non empty set , P1[ XFinSequence] } : ex X being set st for x being set holds ( x in X iff ex p being XFinSequence st ( p in F1() ^omega & P1[p] & x = p ) )
t1_afinsq_1:: for k, n being Nat st k = k /\ n holds k <= n by NAT_1:46;
t10_afinsq_1:: for f being Function for p being XFinSequence st rng p c= dom f holds f * p is XFinSequence
t11_afinsq_1:: for k being Nat for p being XFinSequence st k < len p holds dom (p | k) = k
t12_afinsq_1:: for D being set for f being XFinSequence of D holds f is PartFunc of NAT,D
t13_afinsq_1:: for k being Nat for a being set holds k --> a is XFinSequence ;
t14_afinsq_1:: for k being Nat for D being non empty set ex p being XFinSequence of D st len p = k
t15_afinsq_1:: for p being XFinSequence holds ( len p = 0 iff p = {} ) ;
t16_afinsq_1:: for D being set holds {} is XFinSequence of D
t17_afinsq_1:: for p, q being XFinSequence holds len (p ^ q) = (len p) + (len q) by Def3;
t18_afinsq_1:: for k being Nat for p, q being XFinSequence st len p <= k & k < (len p) + (len q) holds (p ^ q) . k = q . (k - (len p))
t19_afinsq_1:: for k being Nat for p, q being XFinSequence st len p <= k & k < len (p ^ q) holds (p ^ q) . k = q . (k - (len p))
t2_afinsq_1:: for n being Nat holds n \/ {n} = n + 1
t20_afinsq_1:: for k being Nat for p, q being XFinSequence holds ( not k in dom (p ^ q) or k in dom p or ex n being Nat st ( n in dom q & k = (len p) + n ) )
t21_afinsq_1:: for p, q being T-Sequence holds dom p c= dom (p ^ q)
t22_afinsq_1:: for x being set for q, p being XFinSequence st x in dom q holds ex k being Nat st ( k = x & (len p) + k in dom (p ^ q) )
t23_afinsq_1:: for k being Nat for q, p being XFinSequence st k in dom q holds (len p) + k in dom (p ^ q)
t24_afinsq_1:: for p, q being XFinSequence holds rng p c= rng (p ^ q)
t25_afinsq_1:: for q, p being XFinSequence holds rng q c= rng (p ^ q)
t26_afinsq_1:: for p, q being XFinSequence holds rng (p ^ q) = (rng p) \/ (rng q)
t27_afinsq_1:: for p, q, r being XFinSequence holds (p ^ q) ^ r = p ^ (q ^ r)
t28_afinsq_1:: for p, r, q being XFinSequence st ( p ^ r = q ^ r or r ^ p = r ^ q ) holds p = q
t29_afinsq_1:: canceled;
t3_afinsq_1:: for n being Nat holds Seg n c= n + 1
t30_afinsq_1:: for p, q being XFinSequence st p ^ q = {} holds ( p = {} & q = {} )
t31_afinsq_1:: for p, q being XFinSequence for D being set st p ^ q is XFinSequence of D holds ( p is XFinSequence of D & q is XFinSequence of D )
t32_afinsq_1:: for x being set holds <%x%> = {[0,x]} by FUNCT_4:82;
t33_afinsq_1:: for x being set for p being XFinSequence holds ( p = <%x%> iff ( dom p = 1 & rng p = {x} ) )
t34_afinsq_1:: for x being set for p being XFinSequence holds ( p = <%x%> iff ( len p = 1 & p . 0 = x ) ) by Def4;
t35_afinsq_1:: for x being set for p being XFinSequence holds (<%x%> ^ p) . 0 = x
t36_afinsq_1:: for x being set for p being XFinSequence holds (p ^ <%x%>) . (len p) = x
t37_afinsq_1:: for x, y, z being set holds ( <%x,y,z%> = <%x%> ^ <%y,z%> & <%x,y,z%> = <%x,y%> ^ <%z%> ) by Th27;
t38_afinsq_1:: for x, y being set for p being XFinSequence holds ( p = <%x,y%> iff ( len p = 2 & p . 0 = x & p . 1 = y ) )
t39_afinsq_1:: for x, y, z being set for p being XFinSequence holds ( p = <%x,y,z%> iff ( len p = 3 & p . 0 = x & p . 1 = y & p . 2 = z ) )
t4_afinsq_1:: for n being Nat holds n + 1 = {0} \/ (Seg n)
t40_afinsq_1:: for p being XFinSequence st p <> {} holds ex q being XFinSequence ex x being set st p = q ^ <%x%>
t41_afinsq_1:: for p, q, r, s being XFinSequence st p ^ q = r ^ s & len p <= len r holds ex t being XFinSequence st p ^ t = r
t42_afinsq_1:: for x, D being set holds ( x in D ^omega iff x is XFinSequence of D ) by Def7;
t43_afinsq_1:: for D being set holds {} in D ^omega
t44_afinsq_1:: for p being XFinSequence for i being Element of NAT for x being set holds ( len (Replace (p,i,x)) = len p & ( i < len p implies (Replace (p,i,x)) . i = x ) & ( for j being Element of NAT st j <> i holds (Replace (p,i,x)) . j = p . j ) )
t45_afinsq_1:: for p being XFinSequence for x1, x2, x3, x4 being set st p = ((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%> holds ( len p = 4 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 )
t46_afinsq_1:: for p being XFinSequence for x1, x2, x3, x4, x5 being set st p = (((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%> holds ( len p = 5 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 )
t47_afinsq_1:: for p being XFinSequence for x1, x2, x3, x4, x5, x6 being set st p = ((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%> holds ( len p = 6 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 )
t48_afinsq_1:: for p being XFinSequence for x1, x2, x3, x4, x5, x6, x7 being set st p = (((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) ^ <%x7%> holds ( len p = 7 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 & p . 6 = x7 )
t49_afinsq_1:: for p being XFinSequence for x1, x2, x3, x4, x5, x6, x7, x8 being set st p = ((((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) ^ <%x7%>) ^ <%x8%> holds ( len p = 8 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 & p . 6 = x7 & p . 7 = x8 )
t5_afinsq_1:: for r being Function holds ( ( r is finite & r is T-Sequence-like ) iff ex n being Nat st dom r = n ) by FINSET_1:10, ORDINAL1:def_7;
t50_afinsq_1:: for p being XFinSequence for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set st p = (((((((<%x1%> ^ <%x2%>) ^ <%x3%>) ^ <%x4%>) ^ <%x5%>) ^ <%x6%>) ^ <%x7%>) ^ <%x8%>) ^ <%x9%> holds ( len p = 9 & p . 0 = x1 & p . 1 = x2 & p . 2 = x3 & p . 3 = x4 & p . 4 = x5 & p . 5 = x6 & p . 6 = x7 & p . 7 = x8 & p . 8 = x9 )
t51_afinsq_1:: for n being Nat for p, q being XFinSequence st n < len p holds (p ^ q) . n = p . n
t52_afinsq_1:: for n being Nat for p being XFinSequence st len p <= n holds p | n = p
t53_afinsq_1:: for n, k being Nat for p being XFinSequence st n <= len p & k in n holds ( (p | n) . k = p . k & k in dom p )
t54_afinsq_1:: for n being Nat for p being XFinSequence st n <= len p holds len (p | n) = n
t55_afinsq_1:: for n being Nat for p being XFinSequence holds len (p | n) <= n
t56_afinsq_1:: for n being Nat for p being XFinSequence st len p = n + 1 holds p = (p | n) ^ <%(p . n)%>
t57_afinsq_1:: for p, q being XFinSequence holds (p ^ q) | (dom p) = p
t58_afinsq_1:: for n being Nat for p, q being XFinSequence st n <= dom p holds (p ^ q) | n = p | n
t59_afinsq_1:: for n, k being Nat for p, q being XFinSequence st n = (dom p) + k holds (p ^ q) | n = p ^ (q | k)
t6_afinsq_1:: for f being Function st ex k being Nat st dom f c= k holds ex p being XFinSequence st f c= p
t60_afinsq_1:: for n being Nat for p being XFinSequence ex q being XFinSequence st p = (p | n) ^ q
t61_afinsq_1:: for n, k being Nat for p being XFinSequence st len p = n + k holds ex q1, q2 being XFinSequence st ( len q1 = n & len q2 = k & p = q1 ^ q2 )
t62_afinsq_1:: for x, y being set for p, q being XFinSequence st <%x%> ^ p = <%y%> ^ q holds ( x = y & p = q )
t63_afinsq_1:: for D being set for n being Nat for r being set st r in D holds n --> r is XFinSequence of D ;
t64_afinsq_1:: for D being non empty set for p being XFinSequence of D st p . 0 = 0 & 0 < len p holds XFS2FS* p = {}
t65_afinsq_1:: for F being NAT -defined non empty initial Function holds 0 in dom F
t66_afinsq_1:: for F being NAT -defined finite initial Function for n being Nat holds ( n in dom F iff n < card F ) by NAT_1:44;
t67_afinsq_1:: for F being NAT -defined initial Function for G being NAT -defined Function st dom F = dom G holds G is initial
t68_afinsq_1:: for F being NAT -defined finite initial Function holds dom F = { k where k is Element of NAT : k < card F }
t69_afinsq_1:: for F being NAT -defined non empty finite initial Function for G being NAT -defined non empty finite Function st F c= G & LastLoc F = LastLoc G holds F = G
t7_afinsq_1:: for z being set for p being XFinSequence st z in p holds ex k being Nat st ( k in dom p & z = [k,(p . k)] )
t70_afinsq_1:: for F being NAT -defined non empty finite initial Function holds LastLoc F = (card F) -' 1
t71_afinsq_1:: for F being NAT -defined non empty finite initial Function holds FirstLoc F = 0 by Th65, VALUED_1:35;
t72_afinsq_1:: for I being NAT -defined finite initial Function for J being Function holds dom I misses dom (Shift (J,(card I)))
t73_afinsq_1:: for p being XFinSequence for m being Nat st not m in dom p holds not succ m in dom p
t74_afinsq_1:: for p, q being XFinSequence holds p c= p ^ q
t75_afinsq_1:: for x being set for p being XFinSequence holds len (p ^ <%x%>) = (len p) + 1
t76_afinsq_1:: for x, y being set holds <%x,y%> = (0,1) --> (x,y)
t77_afinsq_1:: for p, q being XFinSequence holds p ^ q = p +* (Shift (q,(card p)))
t78_afinsq_1:: for p, q being XFinSequence holds ( p +* (p ^ q) = p ^ q & (p ^ q) +* p = p ^ q )
t79_afinsq_1:: for n being Element of NAT for I being NAT -defined finite initial Function for J being Function holds dom (Shift (I,n)) misses dom (Shift (J,(n + (card I))))
t8_afinsq_1:: for p, q being XFinSequence st dom p = dom q & ( for k being Nat st k in dom p holds p . k = q . k ) holds p = q
t80_afinsq_1:: for p, q being XFinSequence for n being Element of NAT holds Shift (p,n) c= Shift ((p ^ q),n)
t81_afinsq_1:: for q, p being XFinSequence for n being Element of NAT holds Shift (q,(n + (card p))) c= Shift ((p ^ q),n)
t82_afinsq_1:: for X being set for p, q being XFinSequence for n being Element of NAT st Shift ((p ^ q),n) c= X holds Shift (p,n) c= X
t83_afinsq_1:: for X being set for p, q being XFinSequence for n being Element of NAT st Shift ((p ^ q),n) c= X holds Shift (q,(n + (card p))) c= X
t84_afinsq_1:: for x1, x2, x3, x4 being set holds len <%x1,x2,x3,x4%> = 4
t85_afinsq_1:: for x1, x2, x3, x4 being set holds ( <%x1,x2,x3,x4%> . 0 = x1 & <%x1,x2,x3,x4%> . 1 = x2 & <%x1,x2,x3,x4%> . 2 = x3 & <%x1,x2,x3,x4%> . 3 = x4 )
t9_afinsq_1:: for p, q being XFinSequence st len p = len q & ( for k being Nat st k < len p holds p . k = q . k ) holds p = q
d1_afinsq_2:: for p, b2 being XFinSequence holds ( b2 = Rev p iff ( len b2 = len p & ( for i being Nat st i in dom b2 holds b2 . i = p . ((len p) - (i + 1)) ) ) );
d10_afinsq_2:: for D being set for F being XFinSequence of D ^omega for b3 being Element of D ^omega holds ( b3 = FlattenSeq F iff ex g being BinOp of (D ^omega) st ( ( for p, q being Element of D ^omega holds g . (p,q) = p ^ q ) & b3 = g "**" F ) );
d2_afinsq_2:: for p being XFinSequence for n being Nat for b3 being XFinSequence holds ( b3 = p /^ n iff ( len b3 = (len p) -' n & ( for m being Nat st m in dom b3 holds b3 . m = p . (m + n) ) ) );
d3_afinsq_2:: for p being XFinSequence for k1, k2 being Nat holds mid (p,k1,k2) = (p | k2) /^ (k1 -' 1);
d4_afinsq_2:: for X being finite natural-membered set for b2 being XFinSequence of NAT holds ( b2 = Sgm0 X iff ( rng b2 = X & ( for l, m, k1, k2 being Nat st l < m & m < len b2 & k1 = b2 . l & k2 = b2 . m holds k1 < k2 ) ) );
d5_afinsq_2:: for B1, B2 being set holds ( B1 = 1 ) holds for b4 being Element of D holds ( ( b is having_a_unity & len F = 0 implies ( b4 = b "**" F iff b4 = the_unity_wrt b ) ) & ( ( not b is having_a_unity or not len F = 0 ) implies ( b4 = b "**" F iff ex f being Function of NAT,D st ( f . 0 = F . 0 & ( for n being Nat st n + 1 < len F holds f . (n + 1) = b . ((f . n),(F . (n + 1))) ) & b4 = f . ((len F) - 1) ) ) ) );
d9_afinsq_2:: for F being XFinSequence holds Sum F = addcomplex "**" F;
s1_afinsq_2:: scheme XSeqLambdaD{ F1() -> Nat, F2() -> non empty set , F3( set ) -> Element of F2() } : ex p being XFinSequence of F2() st ( len p = F1() & ( for j being Nat st j in F1() holds p . j = F3(j) ) )
s2_afinsq_2:: scheme Sch5{ F1() -> set , P1[ set ] } : for F being XFinSequence of F1() holds P1[F] provided A1: P1[ <%> F1()] and A2: for F being XFinSequence of F1() for d being Element of F1() st P1[F] holds P1[F ^ <%d%>]
t1_afinsq_2:: for i being Nat for x being set st x in i holds x is Element of NAT
t10_afinsq_2:: for p being XFinSequence holds p /^ 0 = p
t11_afinsq_2:: for i being Nat for p, q being XFinSequence holds (p ^ q) /^ ((len p) + i) = q /^ i
t12_afinsq_2:: for p, q being XFinSequence holds (p ^ q) /^ (len p) = q
t13_afinsq_2:: for n being Nat for p being XFinSequence holds (p | n) ^ (p /^ n) = p
t14_afinsq_2:: for p being XFinSequence for k1, k2 being Nat st k1 > k2 holds mid (p,k1,k2) = {}
t15_afinsq_2:: for p being XFinSequence for k1, k2 being Nat st 1 <= k1 & k2 <= len p holds mid (p,k1,k2) = (p /^ (k1 -' 1)) | ((k2 + 1) -' k1)
t16_afinsq_2:: for k being Nat for p being XFinSequence holds mid (p,1,k) = p | k
t17_afinsq_2:: for k being Nat for p being XFinSequence st len p <= k holds mid (p,1,k) = p
t18_afinsq_2:: for k being Nat for p being XFinSequence holds mid (p,0,k) = mid (p,1,k)
t19_afinsq_2:: for p, q being XFinSequence holds mid ((p ^ q),((len p) + 1),((len p) + (len q))) = q
t2_afinsq_2:: for X0 being finite natural-membered set ex n being Nat st X0 c= n
t20_afinsq_2:: for A being finite natural-membered set holds len (Sgm0 A) = card A
t21_afinsq_2:: for X, Y being finite natural-membered set st X c= Y & X <> {} holds (Sgm0 Y) . 0 <= (Sgm0 X) . 0
t22_afinsq_2:: for n being Nat holds (Sgm0 {n}) . 0 = n
t23_afinsq_2:: for B1, B2 being set st B1 {} & ex x being set st ( x in X & {x} {} & X {} & X = d
t38_afinsq_2:: for D being non empty set for b being BinOp of D for d1, d2 being Element of D holds b "**" <%d1,d2%> = b . (d1,d2)
t39_afinsq_2:: for D being non empty set for b being BinOp of D for d, d1, d2 being Element of D holds b "**" <%d,d1,d2%> = b . ((b . (d,d1)),d2)
t4_afinsq_2:: for D being set for p being XFinSequence st ( for i being Nat st i in dom p holds p . i in D ) holds p is XFinSequence of D
t40_afinsq_2:: for D being non empty set for F being XFinSequence of D for b being BinOp of D for d being Element of D st ( b is having_a_unity or len F > 0 ) holds b "**" (F ^ <%d%>) = b . ((b "**" F),d)
t41_afinsq_2:: for D being non empty set for F, G being XFinSequence of D for b being BinOp of D st b is associative & ( b is having_a_unity or ( len F >= 1 & len G >= 1 ) ) holds b "**" (F ^ G) = b . ((b "**" F),(b "**" G))
t42_afinsq_2:: for D being non empty set for F, G being XFinSequence of D for b being BinOp of D st b is associative & ( b is having_a_unity or ( len F >= 1 & len G >= 1 ) ) holds b "**" (F ^ G) = b . ((b "**" F),(b "**" G))
t43_afinsq_2:: for n being Nat for D being non empty set for F being XFinSequence of D for b being BinOp of D st n in dom F & ( b is having_a_unity or n <> 0 ) holds b . ((b "**" (F | n)),(F . n)) = b "**" (F | (n + 1))
t44_afinsq_2:: for D being non empty set for F being XFinSequence of D for b being BinOp of D st ( b is having_a_unity or len F >= 1 ) holds b "**" F = b "**" (XFS2FS F)
t45_afinsq_2:: for D being non empty set for F, G being XFinSequence of D for b being BinOp of D for P being Permutation of (dom F) st b is commutative & b is associative & ( b is having_a_unity or len F >= 1 ) & G = F * P holds b "**" F = b "**" G
t46_afinsq_2:: for D being non empty set for F, G being XFinSequence of D for b being BinOp of D for bFG being XFinSequence of D st b is commutative & b is associative & ( b is having_a_unity or len F >= 1 ) & len F = len G & len F = len bFG & ( for n being Nat st n in dom bFG holds bFG . n = b . ((F . n),(G . n)) ) holds b "**" (F ^ G) = b "**" bFG
t47_afinsq_2:: for D being non empty set for F being XFinSequence of D for D1, D2 being non empty set for b1 being BinOp of D1 for b2 being BinOp of D2 st len F >= 1 & D c= D1 /\ D2 & ( for x, y being set st x in D & y in D holds ( b1 . (x,y) = b2 . (x,y) & b1 . (x,y) in D ) ) holds b1 "**" F = b2 "**" F
t48_afinsq_2:: for F being XFinSequence st F is real-valued holds Sum F = addreal "**" F
t49_afinsq_2:: for F being XFinSequence st F is RAT -valued holds Sum F = addrat "**" F
t5_afinsq_2:: for p being XFinSequence holds ( dom p = dom (Rev p) & rng p = rng (Rev p) )
t50_afinsq_2:: for F being XFinSequence st F is INT -valued holds Sum F = addint "**" F
t51_afinsq_2:: for F being XFinSequence st F is natural-valued holds Sum F = addnat "**" F
t52_afinsq_2:: for cF being complex-valued XFinSequence st cF = {} holds Sum cF = 0 ;
t53_afinsq_2:: for c being complex number holds Sum <%c%> = c
t54_afinsq_2:: for c1, c2 being complex number holds Sum <%c1,c2%> = c1 + c2
t55_afinsq_2:: for cF1, cF2 being complex-valued XFinSequence holds Sum (cF1 ^ cF2) = (Sum cF1) + (Sum cF2)
t56_afinsq_2:: for n being Nat for rF being real-valued XFinSequence for S being Real_Sequence st rF = S | (n + 1) holds Sum rF = (Partial_Sums S) . n
t57_afinsq_2:: for rF1, rF2 being real-valued XFinSequence st len rF1 = len rF2 & ( for i being Nat st i in dom rF1 holds rF1 . i <= rF2 . i ) holds Sum rF1 <= Sum rF2
t58_afinsq_2:: for n being Nat for c being complex number holds Sum (n --> c) = n * c
t59_afinsq_2:: for rF being real-valued XFinSequence for r being real number st ( for n being Nat st n in dom rF holds rF . n <= r ) holds Sum rF <= (len rF) * r
t6_afinsq_2:: for n being Nat for p being XFinSequence st n >= len p holds p /^ n = {}
t60_afinsq_2:: for rF being real-valued XFinSequence for r being real number st ( for n being Nat st n in dom rF holds rF . n >= r ) holds Sum rF >= (len rF) * r
t61_afinsq_2:: for rF being real-valued XFinSequence for r being real number st rF is nonnegative-yielding & len rF > 0 & ex x being set st ( x in dom rF & rF . x = r ) holds Sum rF >= r
t62_afinsq_2:: for rF being real-valued XFinSequence st rF is nonnegative-yielding holds ( Sum rF = 0 iff ( len rF = 0 or rF = (len rF) --> 0 ) )
t63_afinsq_2:: for n being Nat for cF being complex-valued XFinSequence for c being complex number holds c (#) (cF | n) = (c (#) cF) | n
t64_afinsq_2:: for cF being complex-valued XFinSequence for c being complex number holds c * (Sum cF) = Sum (c (#) cF)
t65_afinsq_2:: for n being Nat for cF being complex-valued XFinSequence st n in dom cF holds (Sum (cF | n)) + (cF . n) = Sum (cF | (n + 1))
t66_afinsq_2:: for y, x being set for f being Function st f . y = x & y in dom f holds {y} \/ ((f | ((dom f) \ {y})) " {x}) = f " {x}
t67_afinsq_2:: for y, x being set for f being Function st f . y <> x holds (f | ((dom f) \ {y})) " {x} = f " {x}
t68_afinsq_2:: for cF being complex-valued XFinSequence for c being complex number st rng cF c= {0,c} holds Sum cF = c * (card (cF " {c}))
t69_afinsq_2:: for cF being complex-valued XFinSequence holds Sum cF = Sum (Rev cF)
t7_afinsq_2:: for n being Nat for p being XFinSequence st n < len p holds len (p /^ n) = (len p) - n
t70_afinsq_2:: for f being Function for p, q, fp, fq being XFinSequence st rng p c= dom f & rng q c= dom f & fp = f * p & fq = f * q holds fp ^ fq = f * (p ^ q)
t71_afinsq_2:: for cF being complex-valued XFinSequence for B1, B2 being finite natural-membered set st B1 D) = the_unity_wrt b
t73_afinsq_2:: for D being set for d being Element of D ^omega holds FlattenSeq <%d%> = d
t74_afinsq_2:: for D being set holds FlattenSeq (<%> (D ^omega)) = <%> D
t75_afinsq_2:: for D being set for F, G being XFinSequence of D ^omega holds FlattenSeq (F ^ G) = (FlattenSeq F) ^ (FlattenSeq G)
t76_afinsq_2:: for D being set for p, q being Element of D ^omega holds FlattenSeq <%p,q%> = p ^ q
t77_afinsq_2:: for D being set for p, q, r being Element of D ^omega holds FlattenSeq <%p,q,r%> = (p ^ q) ^ r
t78_afinsq_2:: for p, q being XFinSequence st p c= q holds p ^ (q /^ (len p)) = q
t79_afinsq_2:: for p, q being XFinSequence st p c= q holds ex r being XFinSequence st p ^ r = q
t8_afinsq_2:: for m, n being Nat for p being XFinSequence st m + n < len p holds (p /^ n) . m = p . (m + n)
t80_afinsq_2:: for D being non empty set for p, q being XFinSequence of D st p c= q holds ex r being XFinSequence of D st p ^ r = q
t81_afinsq_2:: for q, p, r being XFinSequence st q c= r holds p ^ q c= p ^ r
t82_afinsq_2:: for D being set for F, G being XFinSequence of D ^omega st F c= G holds FlattenSeq F c= FlattenSeq G
t83_afinsq_2:: for x being set for p being XFinSequence holds CutLastLoc (p ^ <%x%>) = p
t9_afinsq_2:: for n being Nat for p being XFinSequence holds rng (p /^ n) c= rng p
d1_afproj:: for AS being AffinSpace holds AfLines AS = { A where A is Subset of AS : A is being_line } ;
d10_afproj:: for AS being AffinSpace holds ProjectiveLines AS = [:(AfLines AS),{1}:] \/ [:(Dir_of_Planes AS),{2}:];
d11_afproj:: for AS being AffinSpace for b2 being Relation of (ProjectivePoints AS),(ProjectiveLines AS) holds ( b2 = Proj_Inc AS iff for x, y being set holds ( [x,y] in b2 iff ( ex K being Subset of AS st ( K is being_line & y = [K,1] & ( ( x in the carrier of AS & x in K ) or x = LDir K ) ) or ex K, X being Subset of AS st ( K is being_line & X is being_plane & x = LDir K & y = [(PDir X),2] & K '||' X ) ) ) );
d12_afproj:: for AS being AffinSpace for b2 being Relation of (Dir_of_Lines AS),(Dir_of_Planes AS) holds ( b2 = Inc_of_Dir AS iff for x, y being set holds ( [x,y] in b2 iff ex A, X being Subset of AS st ( x = LDir A & y = PDir X & A is being_line & X is being_plane & A '||' X ) ) );
d13_afproj:: for AS being AffinSpace holds IncProjSp_of AS = IncProjStr(# (ProjectivePoints AS),(ProjectiveLines AS),(Proj_Inc AS) #);
d14_afproj:: for AS being AffinSpace holds ProjHorizon AS = IncProjStr(# (Dir_of_Lines AS),(Dir_of_Planes AS),(Inc_of_Dir AS) #);
d2_afproj:: for AS being AffinSpace holds AfPlanes AS = { A where A is Subset of AS : A is being_plane } ;
d3_afproj:: for AS being AffinSpace holds LinesParallelity AS = { [K,M] where K, M is Subset of AS : ( K is being_line & M is being_line & K '||' M ) } ;
d4_afproj:: for AS being AffinSpace holds PlanesParallelity AS = { [X,Y] where X, Y is Subset of AS : ( X is being_plane & Y is being_plane & X '||' Y ) } ;
d5_afproj:: for AS being AffinSpace for X being Subset of AS holds LDir X = Class ((LinesParallelity AS),X);
d6_afproj:: for AS being AffinSpace for X being Subset of AS holds PDir X = Class ((PlanesParallelity AS),X);
d7_afproj:: for AS being AffinSpace holds Dir_of_Lines AS = Class (LinesParallelity AS);
d8_afproj:: for AS being AffinSpace holds Dir_of_Planes AS = Class (PlanesParallelity AS);
d9_afproj:: for AS being AffinSpace holds ProjectivePoints AS = the carrier of AS \/ (Dir_of_Lines AS);
t1_afproj:: for AS being AffinSpace for X being Subset of AS st AS is AffinPlane & X = the carrier of AS holds X is being_plane
t10_afproj:: for AS being AffinSpace for X being Subset of AS st X is being_plane holds for x being set holds ( x in PDir X iff ex Y being Subset of AS st ( x = Y & Y is being_plane & X '||' Y ) )
t11_afproj:: for AS being AffinSpace for X, Y being Subset of AS st X is being_line & Y is being_line holds ( LDir X = LDir Y iff X // Y )
t12_afproj:: for AS being AffinSpace for X, Y being Subset of AS st X is being_line & Y is being_line holds ( LDir X = LDir Y iff X '||' Y )
t13_afproj:: for AS being AffinSpace for X, Y being Subset of AS st X is being_plane & Y is being_plane holds ( PDir X = PDir Y iff X '||' Y )
t14_afproj:: for AS being AffinSpace for x being set holds ( x in Dir_of_Lines AS iff ex X being Subset of AS st ( x = LDir X & X is being_line ) )
t15_afproj:: for AS being AffinSpace for x being set holds ( x in Dir_of_Planes AS iff ex X being Subset of AS st ( x = PDir X & X is being_plane ) )
t16_afproj:: for AS being AffinSpace holds the carrier of AS misses Dir_of_Lines AS
t17_afproj:: for AS being AffinSpace st AS is AffinPlane holds AfLines AS misses Dir_of_Planes AS
t18_afproj:: for AS being AffinSpace for x being set holds ( x in [:(AfLines AS),{1}:] iff ex X being Subset of AS st ( x = [X,1] & X is being_line ) )
t19_afproj:: for AS being AffinSpace for x being set holds ( x in [:(Dir_of_Planes AS),{2}:] iff ex X being Subset of AS st ( x = [(PDir X),2] & X is being_plane ) )
t2_afproj:: for AS being AffinSpace for X being Subset of AS st AS is AffinPlane & X is being_plane holds X = the carrier of AS
t20_afproj:: for AS being AffinSpace for x being set holds ( x is POINT of (IncProjSp_of AS) iff ( x is Element of AS or ex X being Subset of AS st ( x = LDir X & X is being_line ) ) )
t21_afproj:: for AS being AffinSpace for x being set holds ( x is Element of the Points of (ProjHorizon AS) iff ex X being Subset of AS st ( x = LDir X & X is being_line ) ) by Th14;
t22_afproj:: for AS being AffinSpace for x being set st x is Element of the Points of (ProjHorizon AS) holds x is POINT of (IncProjSp_of AS)
t23_afproj:: for AS being AffinSpace for x being set holds ( x is LINE of (IncProjSp_of AS) iff ex X being Subset of AS st ( ( x = [X,1] & X is being_line ) or ( x = [(PDir X),2] & X is being_plane ) ) )
t24_afproj:: for AS being AffinSpace for x being set holds ( x is Element of the Lines of (ProjHorizon AS) iff ex X being Subset of AS st ( x = PDir X & X is being_plane ) ) by Th15;
t25_afproj:: for AS being AffinSpace for x being set st x is Element of the Lines of (ProjHorizon AS) holds [x,2] is LINE of (IncProjSp_of AS)
t26_afproj:: for AS being AffinSpace for x being Element of AS for X being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st x = a & [X,1] = A holds ( a on A iff ( X is being_line & x in X ) )
t27_afproj:: for AS being AffinSpace for x being Element of AS for X being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st x = a & [(PDir X),2] = A holds not a on A
t28_afproj:: for AS being AffinSpace for Y, X being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st a = LDir Y & [X,1] = A & Y is being_line & X is being_line holds ( a on A iff Y '||' X )
t29_afproj:: for AS being AffinSpace for Y, X being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st a = LDir Y & A = [(PDir X),2] & Y is being_line & X is being_plane holds ( a on A iff Y '||' X )
t3_afproj:: for AS being AffinSpace for X, Y being Subset of AS st AS is AffinPlane & X is being_plane & Y is being_plane holds X = Y
t30_afproj:: for AS being AffinSpace for X being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st X is being_line & a = LDir X & A = [X,1] holds a on A
t31_afproj:: for AS being AffinSpace for X, Y being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st X is being_line & Y is being_plane & X c= Y & a = LDir X & A = [(PDir Y),2] holds a on A
t32_afproj:: for AS being AffinSpace for Y, X, X9 being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st Y is being_plane & X c= Y & X9 // X & a = LDir X9 & A = [(PDir Y),2] holds a on A
t33_afproj:: for AS being AffinSpace for X being Subset of AS for a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st A = [(PDir X),2] & X is being_plane & a on A holds not a is Element of AS by Th27;
t34_afproj:: for AS being AffinSpace for X being Subset of AS for p being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st A = [X,1] & X is being_line & p on A & p is not Element of AS holds p = LDir X
t35_afproj:: for AS being AffinSpace for X being Subset of AS for p, a being POINT of (IncProjSp_of AS) for A being LINE of (IncProjSp_of AS) st A = [X,1] & X is being_line & p on A & a on A & a <> p & p is not Element of AS holds a is Element of AS
t36_afproj:: for AS being AffinSpace for X, Y being Subset of AS for a being Element of the Points of (ProjHorizon AS) for A being Element of the Lines of (ProjHorizon AS) st a = LDir X & A = PDir Y & X is being_line & Y is being_plane holds ( a on A iff X '||' Y )
t37_afproj:: for AS being AffinSpace for a being Element of the Points of (ProjHorizon AS) for a9 being Element of the Points of (IncProjSp_of AS) for A being Element of the Lines of (ProjHorizon AS) for A9 being LINE of (IncProjSp_of AS) st a9 = a & A9 = [A,2] holds ( a on A iff a9 on A9 )
t38_afproj:: for AS being AffinSpace for a, b being Element of the Points of (ProjHorizon AS) for A, K being Element of the Lines of (ProjHorizon AS) st a on A & a on K & b on A & b on K & not a = b holds A = K
t39_afproj:: for AS being AffinSpace for A being Element of the Lines of (ProjHorizon AS) ex a, b, c being Element of the Points of (ProjHorizon AS) st ( a on A & b on A & c on A & a <> b & b <> c & c <> a )
t4_afproj:: for AS being AffinSpace for X being Subset of AS st X = the carrier of AS & X is being_plane holds AS is AffinPlane
t40_afproj:: for AS being AffinSpace for a, b being Element of the Points of (ProjHorizon AS) ex A being Element of the Lines of (ProjHorizon AS) st ( a on A & b on A )
t41_afproj:: for AS being AffinSpace for x, y being Element of the Points of (ProjHorizon AS) for X being Element of the Lines of (IncProjSp_of AS) st x <> y & [x,X] in the Inc of (IncProjSp_of AS) & [y,X] in the Inc of (IncProjSp_of AS) holds ex Y being Element of the Lines of (ProjHorizon AS) st X = [Y,2]
t42_afproj:: for AS being AffinSpace for x being POINT of (IncProjSp_of AS) for X being Element of the Lines of (ProjHorizon AS) st [x,[X,2]] in the Inc of (IncProjSp_of AS) holds x is Element of the Points of (ProjHorizon AS)
t43_afproj:: for AS being AffinSpace for Y, X, X9 being Subset of AS for P, Q being LINE of (IncProjSp_of AS) st Y is being_plane & X is being_line & X9 is being_line & X c= Y & X9 c= Y & P = [X,1] & Q = [X9,1] holds ex q being POINT of (IncProjSp_of AS) st ( q on P & q on Q )
t44_afproj:: for AS being AffinSpace for a, b, c, d, p being Element of the Points of (ProjHorizon AS) for M, N, P, Q being Element of the Lines of (ProjHorizon AS) st a on M & b on M & c on N & d on N & p on M & p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M <> N holds ex q being Element of the Points of (ProjHorizon AS) st ( q on P & q on Q )
t45_afproj:: for AS being AffinSpace st IncProjSp_of AS is 2-dimensional holds AS is AffinPlane
t46_afproj:: for AS being AffinSpace st AS is not AffinPlane holds ProjHorizon AS is IncProjSp
t47_afproj:: for AS being AffinSpace st ProjHorizon AS is IncProjSp holds not AS is AffinPlane
t48_afproj:: for AS being AffinSpace for M, N being Subset of AS for o, a, b, c, a9, b9, c9 being Element of AS st M is being_line & N is being_line & M <> N & o in M & o in N & o <> b & o <> b9 & o <> c9 & b in M & c in M & a9 in N & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 & ( a = b or b = c or a = c ) holds a,c9 // c,a9
t49_afproj:: for AS being AffinSpace st IncProjSp_of AS is Pappian holds AS is Pappian
t5_afproj:: for AS being AffinSpace for A, K, X, Y being Subset of AS st not A // K & A '||' X & A '||' Y & K '||' X & K '||' Y & A is being_line & K is being_line & X is being_plane & Y is being_plane holds X '||' Y
t50_afproj:: for AS being AffinSpace for A, P, C being Subset of AS for o, a, b, c, a9, b9, c9 being Element of AS st o in A & o in P & o in C & o <> a & o <> b & o <> c & a in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A <> P & A <> C & a,b // a9,b9 & a,c // a9,c9 & ( o = a9 or a = a9 ) holds b,c // b9,c9
t51_afproj:: for AS being AffinSpace st IncProjSp_of AS is Desarguesian holds AS is Desarguesian
t52_afproj:: for AS being AffinSpace st IncProjSp_of AS is Fanoian holds AS is Fanoian
t6_afproj:: for AS being AffinSpace for X, A, Y being Subset of AS st X is being_plane & A '||' X & X '||' Y holds A '||' Y by AFF_4:59, AFF_4:60;
t7_afproj:: for AS being AffinSpace for x being set holds ( x in AfLines AS iff ex X being Subset of AS st ( x = X & X is being_line ) ) ;
t8_afproj:: for AS being AffinSpace for x being set holds ( x in AfPlanes AS iff ex X being Subset of AS st ( x = X & X is being_plane ) ) ;
t9_afproj:: for AS being AffinSpace for X being Subset of AS st X is being_line holds for x being set holds ( x in LDir X iff ex Y being Subset of AS st ( x = Y & Y is being_line & X '||' Y ) )
d1_afvect0:: for IT being non empty AffinStruct holds ( IT is WeakAffVect-like iff ( ( for a, b, c being Element of IT st a,b // c,c holds a = b ) & ( for a, b, c, d, p, q being Element of IT st a,b // p,q & c,d // p,q holds a,b // c,d ) & ( for a, b, c being Element of IT ex d being Element of IT st a,b // c,d ) & ( for a, b, c, a9, b9, c9 being Element of IT st a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9 ) & ( for a, c being Element of IT ex b being Element of IT st a,b // b,c ) & ( for a, b, c, d being Element of IT st a,b // c,d holds a,c // b,d ) ) );
d10_afvect0:: for X, Y being non empty addLoopStr holds ( X,Y are_Iso iff ex f being Function of the carrier of X, the carrier of Y st f is_Iso_of X,Y );
d2_afvect0:: for AFV being WeakAffVect for a, b being Element of AFV holds ( MDist a,b iff ( a,b // b,a & a <> b ) );
d3_afvect0:: for AFV being WeakAffVect for a, b, c being Element of AFV holds ( Mid a,b,c iff a,b // b,c );
d4_afvect0:: for AFV being WeakAffVect for a, b, b4 being Element of AFV holds ( b4 = PSym (a,b) iff Mid b,a,b4 );
d5_afvect0:: for AFV being WeakAffVect for o, a, b, b5 being Element of AFV holds ( b5 = Padd (o,a,b) iff o,a // b,b5 );
d6_afvect0:: for AFV being WeakAffVect for o being Element of AFV for b3 being BinOp of the carrier of AFV holds ( b3 = Padd o iff for a, b being Element of AFV holds b3 . (a,b) = Padd (o,a,b) );
d7_afvect0:: for AFV being WeakAffVect for o being Element of AFV for b3 being UnOp of the carrier of AFV holds ( b3 = Pcom o iff for a being Element of AFV holds b3 . a = Pcom (o,a) );
d8_afvect0:: for AFV being WeakAffVect for o being Element of AFV holds GroupVect (AFV,o) = addLoopStr(# the carrier of AFV,(Padd o),o #);
d9_afvect0:: for X, Y being non empty addLoopStr for f being Function of the carrier of X, the carrier of Y holds ( f is_Iso_of X,Y iff ( f is one-to-one & rng f = the carrier of Y & ( for a, b being Element of X holds ( f . (a + b) = (f . a) + (f . b) & f . (0. X) = 0. Y & f . (- a) = - (f . a) ) ) ) );
t1_afvect0:: for AFV being WeakAffVect for a, b being Element of AFV holds a,b // a,b
t10_afvect0:: for AFV being WeakAffVect for a, b, a9, b9, c, d, d9 being Element of AFV st a,b // a9,b9 & c,d // b,a & c,d9 // b9,a9 holds d = d9
t11_afvect0:: for AFV being WeakAffVect for a, b, a9, b9, c, d, c9, d9, f, f9 being Element of AFV st a,b // a9,b9 & c,d // c9,d9 & b,f // c,d & b9,f9 // c9,d9 holds a,f // a9,f9
t12_afvect0:: for AFV being WeakAffVect for a, b, a9, b9, c, c9 being Element of AFV st a,b // a9,b9 & a,c // c9,b9 holds b,c // c9,a9
t13_afvect0:: for AFV being WeakAffVect ex a, b being Element of AFV st ( a <> b & not MDist a,b )
t14_afvect0:: for AFV being WeakAffVect for a, b, c being Element of AFV st MDist a,b & MDist a,c & not b = c holds MDist b,c
t15_afvect0:: for AFV being WeakAffVect for a, b, c, d being Element of AFV st MDist a,b & a,b // c,d holds MDist c,d
t16_afvect0:: for AFV being WeakAffVect for a, b, c being Element of AFV st Mid a,b,c holds Mid c,b,a
t17_afvect0:: for AFV being WeakAffVect for a, b being Element of AFV holds ( Mid a,b,b iff a = b )
t18_afvect0:: for AFV being WeakAffVect for a, b being Element of AFV holds ( Mid a,b,a iff ( a = b or MDist a,b ) )
t19_afvect0:: for AFV being WeakAffVect for a, c being Element of AFV ex b being Element of AFV st Mid a,b,c
t2_afvect0:: for AFV being WeakAffVect for a being Element of AFV holds a,a // a,a by Th1;
t20_afvect0:: for AFV being WeakAffVect for a, b, c, b9 being Element of AFV st Mid a,b,c & Mid a,b9,c & not b = b9 holds MDist b,b9
t21_afvect0:: for AFV being WeakAffVect for a, b being Element of AFV ex c being Element of AFV st Mid a,b,c
t22_afvect0:: for AFV being WeakAffVect for a, b, c, c9 being Element of AFV st Mid a,b,c & Mid a,b,c9 holds c = c9
t23_afvect0:: for AFV being WeakAffVect for a, b, c, b9 being Element of AFV st Mid a,b,c & MDist b,b9 holds Mid a,b9,c
t24_afvect0:: for AFV being WeakAffVect for a, b, c, b9, c9 being Element of AFV st Mid a,b,c & Mid a,b9,c9 & MDist b,b9 holds c = c9
t25_afvect0:: for AFV being WeakAffVect for a, p, a9, b, b9 being Element of AFV st Mid a,p,a9 & Mid b,p,b9 holds a,b // b9,a9
t26_afvect0:: for AFV being WeakAffVect for a, p, a9, b, q, b9 being Element of AFV st Mid a,p,a9 & Mid b,q,b9 & MDist p,q holds a,b // b9,a9
t27_afvect0:: for AFV being WeakAffVect for p, a, b being Element of AFV holds ( PSym (p,a) = b iff a,p // p,b )
t28_afvect0:: for AFV being WeakAffVect for p, a being Element of AFV holds ( PSym (p,a) = a iff ( a = p or MDist a,p ) )
t29_afvect0:: for AFV being WeakAffVect for p, a being Element of AFV holds PSym (p,(PSym (p,a))) = a
t3_afvect0:: for AFV being WeakAffVect for a, b, c, d being Element of AFV st a,b // c,d holds c,d // a,b
t30_afvect0:: for AFV being WeakAffVect for p, a, b being Element of AFV st PSym (p,a) = PSym (p,b) holds a = b
t31_afvect0:: for AFV being WeakAffVect for p, b being Element of AFV ex a being Element of AFV st PSym (p,a) = b
t32_afvect0:: for AFV being WeakAffVect for a, b, p being Element of AFV holds a,b // PSym (p,b), PSym (p,a)
t33_afvect0:: for AFV being WeakAffVect for a, b, c, d, p being Element of AFV holds ( a,b // c,d iff PSym (p,a), PSym (p,b) // PSym (p,c), PSym (p,d) )
t34_afvect0:: for AFV being WeakAffVect for a, b, p being Element of AFV holds ( MDist a,b iff MDist PSym (p,a), PSym (p,b) )
t35_afvect0:: for AFV being WeakAffVect for a, b, c, p being Element of AFV holds ( Mid a,b,c iff Mid PSym (p,a), PSym (p,b), PSym (p,c) )
t36_afvect0:: for AFV being WeakAffVect for p, a, q being Element of AFV holds ( PSym (p,a) = PSym (q,a) iff ( p = q or MDist p,q ) )
t37_afvect0:: for AFV being WeakAffVect for q, p, a being Element of AFV holds PSym (q,(PSym (p,(PSym (q,a))))) = PSym ((PSym (q,p)),a)
t38_afvect0:: for AFV being WeakAffVect for p, q, a being Element of AFV holds ( PSym (p,(PSym (q,a))) = PSym (q,(PSym (p,a))) iff ( p = q or MDist p,q or MDist q, PSym (p,q) ) )
t39_afvect0:: for AFV being WeakAffVect for p, q, r, a being Element of AFV holds PSym (p,(PSym (q,(PSym (r,a))))) = PSym (r,(PSym (q,(PSym (p,a)))))
t4_afvect0:: for AFV being WeakAffVect for a, b, c being Element of AFV st a,b // a,c holds b = c
t40_afvect0:: for AFV being WeakAffVect for a, b, c, p being Element of AFV ex d being Element of AFV st PSym (a,(PSym (b,(PSym (c,p))))) = PSym (d,p)
t41_afvect0:: for AFV being WeakAffVect for a, p, b being Element of AFV ex c being Element of AFV st PSym (a,(PSym (c,p))) = PSym (c,(PSym (b,p)))
t42_afvect0:: for AFV being WeakAffVect for o being Element of AFV holds ( the carrier of (GroupVect (AFV,o)) = the carrier of AFV & the addF of (GroupVect (AFV,o)) = Padd o & 0. (GroupVect (AFV,o)) = o ) ;
t43_afvect0:: for AFV being WeakAffVect for o being Element of AFV for a, b being Element of (GroupVect (AFV,o)) for a9, b9 being Element of AFV st a = a9 & b = b9 holds a + b = (Padd o) . (a9,b9) ;
t44_afvect0:: for AFV being WeakAffVect for o being Element of AFV for a being Element of (GroupVect (AFV,o)) for a9 being Element of AFV st a = a9 holds - a = (Pcom o) . a9
t45_afvect0:: for AFV being WeakAffVect for o being Element of AFV holds 0. (GroupVect (AFV,o)) = o ;
t46_afvect0:: for AFV being WeakAffVect for o being Element of AFV for a being Element of (GroupVect (AFV,o)) ex b being Element of (GroupVect (AFV,o)) st b + b = a
t47_afvect0:: for AFV being AffVect for o being Element of AFV for a being Element of (GroupVect (AFV,o)) st a + a = 0. (GroupVect (AFV,o)) holds a = 0. (GroupVect (AFV,o))
t48_afvect0:: for AFV being AffVect for o being Element of AFV holds GroupVect (AFV,o) is Proper_Uniquely_Two_Divisible_Group ;
t49_afvect0:: for ADG being Proper_Uniquely_Two_Divisible_Group holds AV ADG is AffVect
t5_afvect0:: for AFV being WeakAffVect for a, b, c, d, d9 being Element of AFV st a,b // c,d & a,b // c,d9 holds d = d9
t50_afvect0:: for AFV being strict AffVect for o being Element of AFV holds AFV = AV (GroupVect (AFV,o))
t51_afvect0:: for AS being strict AffinStruct holds ( AS is AffVect iff ex ADG being Proper_Uniquely_Two_Divisible_Group st AS = AV ADG )
t52_afvect0:: for ADG being Proper_Uniquely_Two_Divisible_Group for f being Function of the carrier of ADG, the carrier of ADG for o9 being Element of ADG for o being Element of (AV ADG) st ( for x being Element of ADG holds f . x = o9 + x ) & o = o9 holds for a, b being Element of ADG holds ( f . (a + b) = (Padd o) . ((f . a),(f . b)) & f . (0. ADG) = 0. (GroupVect ((AV ADG),o)) & f . (- a) = (Pcom o) . (f . a) )
t53_afvect0:: for ADG being Proper_Uniquely_Two_Divisible_Group for f being Function of the carrier of ADG, the carrier of ADG for o9 being Element of ADG st ( for b being Element of ADG holds f . b = o9 + b ) holds f is one-to-one
t54_afvect0:: for ADG being Proper_Uniquely_Two_Divisible_Group for f being Function of the carrier of ADG, the carrier of ADG for o9 being Element of ADG for o being Element of (AV ADG) st ( for b being Element of ADG holds f . b = o9 + b ) holds rng f = the carrier of (GroupVect ((AV ADG),o))
t55_afvect0:: for ADG being Proper_Uniquely_Two_Divisible_Group for o9 being Element of ADG for o being Element of (AV ADG) st o = o9 holds ADG, GroupVect ((AV ADG),o) are_Iso
t6_afvect0:: for AFV being WeakAffVect for a, b being Element of AFV holds a,a // b,b
t7_afvect0:: for AFV being WeakAffVect for a, b, c, d being Element of AFV st a,b // c,d holds b,a // d,c
t8_afvect0:: for AFV being WeakAffVect for a, b, c, d, b9 being Element of AFV st a,b // c,d & a,c // b9,d holds b = b9
t9_afvect0:: for AFV being WeakAffVect for b, c, b9, c9, a, d, d9 being Element of AFV st b,c // b9,c9 & a,d // b,c & a,d9 // b9,c9 holds d = d9
d1_afvect01:: for IT being non empty AffinStruct holds ( IT is WeakAffSegm-like iff ( ( for a, b being Element of IT holds a,b // b,a ) & ( for a, b being Element of IT st a,b // a,a holds a = b ) & ( for a, b, c, d, p, q being Element of IT st a,b // p,q & c,d // p,q holds a,b // c,d ) & ( for a, c being Element of IT ex b being Element of IT st a,b // b,c ) & ( for a, a9, b, b9, p being Element of IT st a <> a9 & b <> b9 & p,a // p,a9 & p,b // p,b9 holds a,b // a9,b9 ) & ( for a, b being Element of IT holds ( a = b or ex c being Element of IT st ( ( a <> c & a,b // b,c ) or ex p, p9 being Element of IT st ( p <> p9 & a,b // p,p9 & a,p // p,b & a,p9 // p9,b ) ) ) ) & ( for a, b, b9, p, p9, c being Element of IT st a,b // b,c & b,b9 // p,p9 & b,p // p,b9 & b,p9 // p9,b9 holds a,b9 // b9,c ) & ( for a, b, b9, c being Element of IT st a <> c & b <> b9 & a,b // b,c & a,b9 // b9,c holds ex p, p9 being Element of IT st ( p <> p9 & b,b9 // p,p9 & b,p // p,b9 & b,p9 // p9,b9 ) ) & ( for a, b, c, p, p9, q, q9 being Element of IT st a,b // p,p9 & a,c // q,q9 & a,p // p,b & a,q // q,c & a,p9 // p9,b & a,q9 // q9,c holds ex r, r9 being Element of IT st ( b,c // r,r9 & b,r // r,c & b,r9 // r9,c ) ) ) );
d2_afvect01:: for AFV being WeakAffSegm for a, b being Element of AFV holds ( MDist a,b iff ex p, p9 being Element of AFV st ( p <> p9 & a,b // p,p9 & a,p // p,b & a,p9 // p9,b ) );
d3_afvect01:: for AFV being WeakAffSegm for a, b, c being Element of AFV holds ( Mid a,b,c iff ( ( a = b & b = c & a = c ) or ( a = c & MDist a,b ) or ( a <> c & a,b // b,c ) ) );
t1_afvect01:: for AFV being WeakAffSegm for a, b being Element of AFV holds a,b // a,b
t10_afvect01:: for AFV being WeakAffSegm for a, b, c being Element of AFV st MDist a,b & a,b // b,c holds a = c
t11_afvect01:: for AFV being WeakAffSegm for a, b being Element of AFV st MDist a,b holds a <> b
t2_afvect01:: for AFV being WeakAffSegm for a, b, c, d being Element of AFV st a,b // c,d holds c,d // a,b
t3_afvect01:: for AFV being WeakAffSegm for a, b, c, d being Element of AFV st a,b // c,d holds a,b // d,c
t4_afvect01:: for AFV being WeakAffSegm for a, b, c, d being Element of AFV st a,b // c,d holds b,a // c,d
t5_afvect01:: for AFV being WeakAffSegm for a, b being Element of AFV holds a,a // b,b
t6_afvect01:: for AFV being WeakAffSegm for a, b, c being Element of AFV st a,b // c,c holds a = b
t7_afvect01:: for AFV being WeakAffSegm for a, b, p, p9, c being Element of AFV st a,b // p,p9 & a,b // b,c & a,p // p,b & a,p9 // p9,b holds a = c
t8_afvect01:: for AFV being WeakAffSegm for a, b9, b99, b being Element of AFV st a,b9 // a,b99 & a,b // a,b99 & not b = b9 & not b = b99 holds b9 = b99
t9_afvect01:: for AFV being WeakAffSegm for a, b being Element of AFV st a <> b & not MDist a,b holds ex c being Element of AFV st ( a <> c & a,b // b,c )
d1_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 holds ( f is_homomorphism U1,U2 iff ( U1,U2 are_similar & ( for n being Element of NAT st n in dom the charact of U1 holds for o1 being operation of U1 for o2 being operation of U2 st o1 = the charact of U1 . n & o2 = the charact of U2 . n holds for x being FinSequence of U1 st x in dom o1 holds f . (o1 . x) = o2 . (f * x) ) ) );
d10_alg_1:: for U1 being Universal_Algebra for E being Congruence of U1 holds QuotUnivAlg (U1,E) = UAStr(# (Class E),(QuotOpSeq (U1,E)) #);
d11_alg_1:: for U1 being Universal_Algebra for E being Congruence of U1 for b3 being Function of U1,(QuotUnivAlg (U1,E)) holds ( b3 = Nat_Hom (U1,E) iff for u being Element of U1 holds b3 . u = Class (E,u) );
d12_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 st f is_homomorphism U1,U2 holds for b4 being Congruence of U1 holds ( b4 = Cng f iff for a, b being Element of U1 holds ( [a,b] in b4 iff f . a = f . b ) );
d13_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 st f is_homomorphism U1,U2 holds for b4 being Function of (QuotUnivAlg (U1,(Cng f))),U2 holds ( b4 = HomQuot f iff for a being Element of U1 holds b4 . (Class ((Cng f),a)) = f . a );
d2_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 holds ( f is_monomorphism U1,U2 iff ( f is_homomorphism U1,U2 & f is one-to-one ) );
d3_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 holds ( f is_epimorphism U1,U2 iff ( f is_homomorphism U1,U2 & rng f = the carrier of U2 ) );
d4_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 holds ( f is_isomorphism U1,U2 iff ( f is_monomorphism U1,U2 & f is_epimorphism U1,U2 ) );
d5_alg_1:: for U1, U2 being Universal_Algebra holds ( U1,U2 are_isomorphic iff ex f being Function of U1,U2 st f is_isomorphism U1,U2 );
d6_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 st f is_homomorphism U1,U2 holds for b4 being strict SubAlgebra of U2 holds ( b4 = Image f iff the carrier of b4 = f .: the carrier of U1 );
d7_alg_1:: for U1 being Universal_Algebra for b2 being Equivalence_Relation of U1 holds ( b2 is Congruence of U1 iff for n being Element of NAT for o1 being operation of U1 st n in dom the charact of U1 & o1 = the charact of U1 . n holds for x, y being FinSequence of U1 st x in dom o1 & y in dom o1 & [x,y] in ExtendRel b2 holds [(o1 . x),(o1 . y)] in b2 );
d8_alg_1:: for U1 being Universal_Algebra for E being Congruence of U1 for o being operation of U1 for b4 being non empty homogeneous quasi_total PartFunc of ((Class E) *),(Class E) holds ( b4 = QuotOp (o,E) iff ( dom b4 = (arity o) -tuples_on (Class E) & ( for y being FinSequence of Class E st y in dom b4 holds for x being FinSequence of the carrier of U1 st x is_representatives_FS y holds b4 . y = Class (E,(o . x)) ) ) );
d9_alg_1:: for U1 being Universal_Algebra for E being Congruence of U1 for b3 being PFuncFinSequence of (Class E) holds ( b3 = QuotOpSeq (U1,E) iff ( len b3 = len the charact of U1 & ( for n being Element of NAT st n in dom b3 holds for o1 being operation of U1 st the charact of U1 . n = o1 holds b3 . n = QuotOp (o1,E) ) ) );
t1_alg_1:: for U1 being Universal_Algebra for B being non empty Subset of U1 st B = the carrier of U1 holds Opers (U1,B) = the charact of U1
t10_alg_1:: for U1, U2 being Universal_Algebra for h being Function of U1,U2 for h1 being Function of U2,U1 st h is_isomorphism U1,U2 & h1 = h " holds h1 is_isomorphism U2,U1
t11_alg_1:: for U1, U2, U3 being Universal_Algebra for h being Function of U1,U2 for h1 being Function of U2,U3 st h is_isomorphism U1,U2 & h1 is_isomorphism U2,U3 holds h1 * h is_isomorphism U1,U3
t12_alg_1:: for U1 being Universal_Algebra holds U1,U1 are_isomorphic
t13_alg_1:: for U1, U2 being Universal_Algebra st U1,U2 are_isomorphic holds U2,U1 are_isomorphic
t14_alg_1:: for U1, U2, U3 being Universal_Algebra st U1,U2 are_isomorphic & U2,U3 are_isomorphic holds U1,U3 are_isomorphic
t15_alg_1:: for U1, U2 being Universal_Algebra for h being Function of U1,U2 st h is_homomorphism U1,U2 holds rng h = the carrier of (Image h)
t16_alg_1:: for U1 being Universal_Algebra for U2 being strict Universal_Algebra for f being Function of U1,U2 st f is_homomorphism U1,U2 holds ( f is_epimorphism U1,U2 iff Image f = U2 )
t17_alg_1:: for U1 being Universal_Algebra for E being Congruence of U1 holds Nat_Hom (U1,E) is_homomorphism U1, QuotUnivAlg (U1,E)
t18_alg_1:: for U1 being Universal_Algebra for E being Congruence of U1 holds Nat_Hom (U1,E) is_epimorphism U1, QuotUnivAlg (U1,E)
t19_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 st f is_homomorphism U1,U2 holds ( HomQuot f is_homomorphism QuotUnivAlg (U1,(Cng f)),U2 & HomQuot f is_monomorphism QuotUnivAlg (U1,(Cng f)),U2 )
t2_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 holds f * (<*> the carrier of U1) = <*> the carrier of U2 ;
t20_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 st f is_epimorphism U1,U2 holds HomQuot f is_isomorphism QuotUnivAlg (U1,(Cng f)),U2
t21_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 st f is_epimorphism U1,U2 holds QuotUnivAlg (U1,(Cng f)),U2 are_isomorphic
t3_alg_1:: for U1 being Universal_Algebra for a being FinSequence of U1 holds (id the carrier of U1) * a = a
t4_alg_1:: for U1, U2, U3 being Universal_Algebra for h1 being Function of U1,U2 for h2 being Function of U2,U3 for a being FinSequence of U1 holds h2 * (h1 * a) = (h2 * h1) * a
t5_alg_1:: for U1 being Universal_Algebra holds id the carrier of U1 is_homomorphism U1,U1
t6_alg_1:: for U1, U2, U3 being Universal_Algebra for h1 being Function of U1,U2 for h2 being Function of U2,U3 st h1 is_homomorphism U1,U2 & h2 is_homomorphism U2,U3 holds h2 * h1 is_homomorphism U1,U3
t7_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 holds ( f is_isomorphism U1,U2 iff ( f is_homomorphism U1,U2 & rng f = the carrier of U2 & f is one-to-one ) )
t8_alg_1:: for U1, U2 being Universal_Algebra for f being Function of U1,U2 st f is_isomorphism U1,U2 holds ( dom f = the carrier of U1 & rng f = the carrier of U2 )
t9_alg_1:: for U1, U2 being Universal_Algebra for h being Function of U1,U2 for h1 being Function of U2,U1 st h is_isomorphism U1,U2 & h1 = h " holds h1 is_homomorphism U2,U1
d1_algseq_1:: for R being non empty ZeroStr for F being sequence of R holds ( F is finite-Support iff ex n being Nat st for i being Nat st i >= n holds F . i = 0. R );
d2_algseq_1:: for R being non empty ZeroStr for p being AlgSequence of R for k being Nat holds ( k is_at_least_length_of p iff for i being Nat st i >= k holds p . i = 0. R );
d3_algseq_1:: for R being non empty ZeroStr for p being AlgSequence of R for b3 being Element of NAT holds ( b3 = len p iff ( b3 is_at_least_length_of p & ( for m being Nat st m is_at_least_length_of p holds b3 <= m ) ) );
d4_algseq_1:: for R being non empty ZeroStr for p being AlgSequence of R holds support p = Segm (len p);
d5_algseq_1:: for R being non empty ZeroStr for x being Element of R for b3 being AlgSequence of R holds ( b3 = <%x%> iff ( len b3 <= 1 & b3 . 0 = x ) );
s1_algseq_1:: scheme AlgSeqLambdaF{ F1() -> non empty ZeroStr , F2() -> Nat, F3( Nat) -> Element of F1() } : ex p being AlgSequence of F1() st ( len p <= F2() & ( for k being Nat st k < F2() holds p . k = F3(k) ) )
t1_algseq_1:: for k, n being Nat holds ( k in Segm n iff k < n ) by NAT_1:44;
t10_algseq_1:: for k being Nat for R being non empty ZeroStr for p being AlgSequence of R st len p = k + 1 holds p . k <> 0. R
t11_algseq_1:: for k being Nat for R being non empty ZeroStr for p being AlgSequence of R holds ( k = len p iff Segm k = support p ) ;
t12_algseq_1:: for R being non empty ZeroStr for p, q being AlgSequence of R st len p = len q & ( for k being Nat st k < len p holds p . k = q . k ) holds p = q
t13_algseq_1:: for R being non empty ZeroStr st the carrier of R <> {(0. R)} holds for k being Nat ex p being AlgSequence of R st len p = k
t14_algseq_1:: for R being non empty ZeroStr for p being AlgSequence of R holds ( p = <%(0. R)%> iff len p = 0 )
t15_algseq_1:: for R being non empty ZeroStr for p being AlgSequence of R holds ( p = <%(0. R)%> iff support p = {} ) by Th14;
t16_algseq_1:: for i being Nat for R being non empty ZeroStr holds <%(0. R)%> . i = 0. R
t17_algseq_1:: for R being non empty ZeroStr for p being AlgSequence of R holds ( p = <%(0. R)%> iff rng p = {(0. R)} )
t2_algseq_1:: ( Segm 0 = {} & Segm 1 = {0} & Segm 2 = {0,1} ) by CARD_1:49, CARD_1:50;
t3_algseq_1:: for n being Nat holds n in Segm (n + 1) by NAT_1:45;
t4_algseq_1:: for n, m being Nat holds ( n <= m iff Segm n c= Segm m ) by NAT_1:39;
t5_algseq_1:: for n, m being Nat st Segm n = Segm m holds n = m ;
t6_algseq_1:: for k, n being Nat st k <= n holds ( Segm k = (Segm k) /\ (Segm n) & Segm k = (Segm n) /\ (Segm k) ) by NAT_1:46;
t7_algseq_1:: for k, n being Nat st ( Segm k = (Segm k) /\ (Segm n) or Segm k = (Segm n) /\ (Segm k) ) holds k <= n by NAT_1:46;
t8_algseq_1:: for i being Nat for R being non empty ZeroStr for p being AlgSequence of R st i >= len p holds p . i = 0. R
t9_algseq_1:: for k being Nat for R being non empty ZeroStr for p being AlgSequence of R st ( for i being Nat st i < k holds p . i <> 0. R ) holds len p >= k
d1_algspec1:: for X being set for f being Function holds X -indexing f = (id X) +* (f | X);
d2_algspec1:: for f, b2 being Function holds ( b2 is rng-retract of f iff ( dom b2 = rng f & f * b2 = id (rng f) ) );
d3_algspec1:: for S being non empty non void ManySortedSign for f, g being Function holds ( f,g form_a_replacement_in S iff for o1, o2 being OperSymbol of S st ((id the carrier' of S) +* g) . o1 = ((id the carrier' of S) +* g) . o2 holds ( ((id the carrier of S) +* f) * (the_arity_of o1) = ((id the carrier of S) +* f) * (the_arity_of o2) & ((id the carrier of S) +* f) . (the_result_sort_of o1) = ((id the carrier of S) +* f) . (the_result_sort_of o2) ) );
d4_algspec1:: for S being non empty non void ManySortedSign for f, g being Function st f,g form_a_replacement_in S holds for b4 being non empty non void strict ManySortedSign holds ( b4 = S with-replacement (f,g) iff ( the carrier of S -indexing f, the carrier' of S -indexing g form_morphism_between S,b4 & the carrier of b4 = rng ( the carrier of S -indexing f) & the carrier' of b4 = rng ( the carrier' of S -indexing g) ) );
d5_algspec1:: for S, b2 being Signature holds ( b2 is Extension of S iff S is Subsignature of b2 );
d6_algspec1:: for b1 being set holds ( b1 is Algebra iff ex S being non void Signature st b1 is feasible MSAlgebra over S );
d7_algspec1:: for S being Signature for b2 being Algebra holds ( b2 is Algebra of S iff ex E being non void Extension of S st b2 is feasible MSAlgebra over E );
t1_algspec1:: for f, g, h being Function st (dom f) /\ (dom g) c= dom h holds (f +* g) +* h = (g +* f) +* h
t10_algspec1:: for X being set for f being Function st dom f = X holds X -indexing f = f
t11_algspec1:: for X being set for f being Function holds X -indexing (X -indexing f) = X -indexing f
t12_algspec1:: for X being set for f being Function holds X -indexing ((id X) +* f) = X -indexing f
t13_algspec1:: for X being set for f being Function st f c= id X holds X -indexing f = id X
t14_algspec1:: for X being set holds X -indexing {} = id X ;
t15_algspec1:: for X being set for f being Function st X c= dom f holds X -indexing f = f | X
t16_algspec1:: for f being Function holds {} -indexing f = {} ;
t17_algspec1:: for X, Y being set for f being Function st X c= Y holds (Y -indexing f) | X = X -indexing f
t18_algspec1:: for X, Y being set for f being Function holds (X \/ Y) -indexing f = (X -indexing f) +* (Y -indexing f)
t19_algspec1:: for X, Y being set for f being Function holds X -indexing f tolerates Y -indexing f
t2_algspec1:: for f, g, h being Function st f c= g & (rng h) /\ (dom g) c= dom f holds g * h = f * h
t20_algspec1:: for X, Y being set for f being Function holds (X \/ Y) -indexing f = (X -indexing f) \/ (Y -indexing f)
t21_algspec1:: for X being non empty set for f, g being Function st rng g c= X holds (X -indexing f) * g = ((id X) +* f) * g
t22_algspec1:: for f, g being Function st dom f misses dom g & rng g misses dom f holds for X being set holds f * (X -indexing g) = f | X
t23_algspec1:: for f being Function for g being rng-retract of f holds rng g c= dom f
t24_algspec1:: for f being Function for g being rng-retract of f for x being set st x in rng f holds ( g . x in dom f & f . (g . x) = x )
t25_algspec1:: for f being Function st f is one-to-one holds f " is rng-retract of f
t26_algspec1:: for f being Function st f is one-to-one holds for g being rng-retract of f holds g = f "
t27_algspec1:: for f1, f2 being Function st f1 tolerates f2 holds for g1 being rng-retract of f1 for g2 being rng-retract of f2 holds g1 +* g2 is rng-retract of f1 +* f2
t28_algspec1:: for f1, f2 being Function st f1 c= f2 holds for g1 being rng-retract of f1 ex g2 being rng-retract of f2 st g1 c= g2
t29_algspec1:: for S being non empty non void ManySortedSign for f, g being Function holds ( f,g form_a_replacement_in S iff for o1, o2 being OperSymbol of S st ( the carrier' of S -indexing g) . o1 = ( the carrier' of S -indexing g) . o2 holds ( ( the carrier of S -indexing f) * (the_arity_of o1) = ( the carrier of S -indexing f) * (the_arity_of o2) & ( the carrier of S -indexing f) . (the_result_sort_of o1) = ( the carrier of S -indexing f) . (the_result_sort_of o2) ) )
t3_algspec1:: for f, g, h being Function st dom f misses rng h & g .: (dom h) misses dom f holds f * (g +* h) = f * g
t30_algspec1:: for S being non empty non void ManySortedSign for f, g being Function holds ( f,g form_a_replacement_in S iff the carrier of S -indexing f, the carrier' of S -indexing g form_a_replacement_in S )
t31_algspec1:: for S, S9 being non void Signature for f, g being Function st f,g form_morphism_between S,S9 holds f,g form_a_replacement_in S
t32_algspec1:: for S being non void Signature for f being Function holds f, {} form_a_replacement_in S
t33_algspec1:: for S being non void Signature for g, f being Function st g is one-to-one & the carrier' of S /\ (rng g) c= dom g holds f,g form_a_replacement_in S
t34_algspec1:: for S being non void Signature for g, f being Function st g is one-to-one & rng g misses the carrier' of S holds f,g form_a_replacement_in S
t35_algspec1:: for S1, S2 being non void Signature for f being Function of the carrier of S1, the carrier of S2 for g being Function st f,g form_morphism_between S1,S2 holds (f *) * the Arity of S1 = the Arity of S2 * g
t36_algspec1:: for S being non void Signature for f, g being Function st f,g form_a_replacement_in S holds the carrier of S -indexing f is Function of the carrier of S, the carrier of (S with-replacement (f,g))
t37_algspec1:: for S being non void Signature for f, g being Function st f,g form_a_replacement_in S holds for f9 being Function of the carrier of S, the carrier of (S with-replacement (f,g)) st f9 = the carrier of S -indexing f holds for g9 being rng-retract of the carrier' of S -indexing g holds the Arity of (S with-replacement (f,g)) = ((f9 *) * the Arity of S) * g9
t38_algspec1:: for S being non void Signature for f, g being Function st f,g form_a_replacement_in S holds for g9 being rng-retract of the carrier' of S -indexing g holds the ResultSort of (S with-replacement (f,g)) = (( the carrier of S -indexing f) * the ResultSort of S) * g9
t39_algspec1:: for S, S9 being non void Signature for f, g being Function st f,g form_morphism_between S,S9 holds S with-replacement (f,g) is Subsignature of S9
t4_algspec1:: for f1, f2, g1, g2 being Function st f1 tolerates f2 & g1 tolerates g2 holds f1 * g1 tolerates f2 * g2
t40_algspec1:: for S being non void Signature for f, g being Function holds ( f,g form_a_replacement_in S iff the carrier of S -indexing f, the carrier' of S -indexing g form_morphism_between S,S with-replacement (f,g) )
t41_algspec1:: for S being non void Signature for f, g being Function st dom f c= the carrier of S & dom g c= the carrier' of S & f,g form_a_replacement_in S holds (id the carrier of S) +* f,(id the carrier' of S) +* g form_morphism_between S,S with-replacement (f,g)
t42_algspec1:: for S being non void Signature for f, g being Function st dom f = the carrier of S & dom g = the carrier' of S & f,g form_a_replacement_in S holds f,g form_morphism_between S,S with-replacement (f,g)
t43_algspec1:: for S being non void Signature for f, g being Function st f,g form_a_replacement_in S holds S with-replacement (( the carrier of S -indexing f),g) = S with-replacement (f,g)
t44_algspec1:: for S being non void Signature for f, g being Function st f,g form_a_replacement_in S holds S with-replacement (f,( the carrier' of S -indexing g)) = S with-replacement (f,g)
t45_algspec1:: for S being Signature holds S is Extension of S
t46_algspec1:: for S1 being Signature for S2 being Extension of S1 for S3 being Extension of S2 holds S3 is Extension of S1
t47_algspec1:: for S1, S2 being non empty Signature st S1 tolerates S2 holds S1 +* S2 is Extension of S1
t48_algspec1:: for S1, S2 being non empty Signature holds S1 +* S2 is Extension of S2
t49_algspec1:: for S1, S2, S being non empty ManySortedSign for f1, g1, f2, g2 being Function st f1 tolerates f2 & f1,g1 form_morphism_between S1,S & f2,g2 form_morphism_between S2,S holds f1 +* f2,g1 +* g2 form_morphism_between S1 +* S2,S
t5_algspec1:: for X1, Y1, X2, Y2 being non empty set for f being Function of X1,X2 for g being Function of Y1,Y2 st f c= g holds f * c= g *
t50_algspec1:: for S1, S2, E being non empty Signature holds ( E is Extension of S1 & E is Extension of S2 iff ( S1 tolerates S2 & E is Extension of S1 +* S2 ) )
t51_algspec1:: for S, T being Signature st S is empty holds T is Extension of S
t52_algspec1:: for f, g being Function for S being non void Signature for E being Extension of S st f,g form_a_replacement_in E holds f,g form_a_replacement_in S
t53_algspec1:: for f, g being Function for S being non void Signature for E being Extension of S st f,g form_a_replacement_in E holds E with-replacement (f,g) is Extension of S with-replacement (f,g)
t54_algspec1:: for S1, S2 being non void Signature st S1 tolerates S2 holds for f, g being Function st f,g form_a_replacement_in S1 +* S2 holds (S1 +* S2) with-replacement (f,g) = (S1 with-replacement (f,g)) +* (S2 with-replacement (f,g))
t55_algspec1:: for S being non void Signature for A being feasible MSAlgebra over S holds A is Algebra of S
t56_algspec1:: for S being Signature for E being Extension of S for A being Algebra of E holds A is Algebra of S
t57_algspec1:: for S being Signature for E being non empty Signature for A being MSAlgebra over E st A is Algebra of S holds ( the carrier of S c= the carrier of E & the carrier' of S c= the carrier' of E )
t58_algspec1:: for S being non void Signature for E being non empty Signature for A being MSAlgebra over E st A is Algebra of S holds for o being OperSymbol of S holds the Charact of A . o is Function of (( the Sorts of A #) . (the_arity_of o)),( the Sorts of A . (the_result_sort_of o))
t59_algspec1:: for S being non empty Signature for A being Algebra of S for E being non empty ManySortedSign st A is MSAlgebra over E holds A is MSAlgebra over E +* S
t6_algspec1:: for X1, Y1, X2, Y2 being non empty set for f being Function of X1,X2 for g being Function of Y1,Y2 st f tolerates g holds f * tolerates g *
t60_algspec1:: for S1, S2 being non empty Signature for A being MSAlgebra over S1 st A is MSAlgebra over S2 holds ( the carrier of S1 = the carrier of S2 & the carrier' of S1 = the carrier' of S2 )
t61_algspec1:: for S being non void Signature for A being disjoint_valued MSAlgebra over S for C1, C2 being Component of the Sorts of A holds ( C1 = C2 or C1 misses C2 )
t62_algspec1:: for S, S9 being non void Signature for A being non-empty disjoint_valued MSAlgebra over S st A is MSAlgebra over S9 holds ManySortedSign(# the carrier of S, the carrier' of S, the Arity of S, the ResultSort of S #) = ManySortedSign(# the carrier of S9, the carrier' of S9, the Arity of S9, the ResultSort of S9 #)
t63_algspec1:: for S, S9 being non void Signature for A being non-empty disjoint_valued MSAlgebra over S st A is Algebra of S9 holds S is Extension of S9
t7_algspec1:: for X being set for f being Function holds rng (X -indexing f) = (X \ (dom f)) \/ (f .: X)
t8_algspec1:: for X being non empty set for f being Function for x being Element of X holds (X -indexing f) . x = ((id X) +* f) . x
t9_algspec1:: for X, x being set for f being Function st x in X holds ( ( x in dom f implies (X -indexing f) . x = f . x ) & ( not x in dom f implies (X -indexing f) . x = x ) )
d1_algstr_0:: for M being addMagma for x, y being Element of M holds x + y = the addF of M . (x,y);
d10_algstr_0:: for M being addLoopStr for x being Element of M holds ( x is left_complementable iff ex y being Element of M st y + x = 0. M );
d11_algstr_0:: for M being addLoopStr for x being Element of M holds ( x is right_complementable iff ex y being Element of M st x + y = 0. M );
d12_algstr_0:: for M being addLoopStr for x being Element of M holds ( x is complementable iff ( x is right_complementable & x is left_complementable ) );
d13_algstr_0:: for M being addLoopStr for x being Element of M st x is left_complementable & x is right_add-cancelable holds for b3 being Element of M holds ( b3 = - x iff b3 + x = 0. M );
d14_algstr_0:: for V being addLoopStr for v, w being Element of V holds v - w = v + (- w);
d15_algstr_0:: for M being addLoopStr holds ( M is left_complementable iff for x being Element of M holds x is left_complementable );
d16_algstr_0:: for M being addLoopStr holds ( M is right_complementable iff for x being Element of M holds x is right_complementable );
d17_algstr_0:: for M being addLoopStr holds ( M is complementable iff ( M is right_complementable & M is left_complementable ) );
d18_algstr_0:: for M being multMagma for x, y being Element of M holds x * y = the multF of M . (x,y);
d19_algstr_0:: Trivial-multMagma = multMagma(# 1,op2 #);
d2_algstr_0:: Trivial-addMagma = addMagma(# 1,op2 #);
d20_algstr_0:: for M being multMagma for x being Element of M holds ( x is left_mult-cancelable iff for y, z being Element of M st x * y = x * z holds y = z );
d21_algstr_0:: for M being multMagma for x being Element of M holds ( x is right_mult-cancelable iff for y, z being Element of M st y * x = z * x holds y = z );
d22_algstr_0:: for M being multMagma for x being Element of M holds ( x is mult-cancelable iff ( x is right_mult-cancelable & x is left_mult-cancelable ) );
d23_algstr_0:: for M being multMagma holds ( M is left_mult-cancelable iff for x being Element of M holds x is left_mult-cancelable );
d24_algstr_0:: for M being multMagma holds ( M is right_mult-cancelable iff for x being Element of M holds x is right_mult-cancelable );
d25_algstr_0:: for M being multMagma holds ( M is mult-cancelable iff ( M is left_mult-cancelable & M is right_mult-cancelable ) );
d26_algstr_0:: Trivial-multLoopStr = multLoopStr(# 1,op2,op0 #);
d27_algstr_0:: for M being multLoopStr for x being Element of M holds ( x is left_invertible iff ex y being Element of M st y * x = 1. M );
d28_algstr_0:: for M being multLoopStr for x being Element of M holds ( x is right_invertible iff ex y being Element of M st x * y = 1. M );
d29_algstr_0:: for M being multLoopStr for x being Element of M holds ( x is invertible iff ( x is right_invertible & x is left_invertible ) );
d3_algstr_0:: for M being addMagma for x being Element of M holds ( x is left_add-cancelable iff for y, z being Element of M st x + y = x + z holds y = z );
d30_algstr_0:: for M being multLoopStr for x being Element of M st x is left_invertible & x is right_mult-cancelable holds for b3 being Element of M holds ( b3 = / x iff b3 * x = 1. M );
d31_algstr_0:: for M being multLoopStr holds ( M is left_invertible iff for x being Element of M holds x is left_invertible );
d32_algstr_0:: for M being multLoopStr holds ( M is right_invertible iff for x being Element of M holds x is right_invertible );
d33_algstr_0:: for M being multLoopStr holds ( M is invertible iff ( M is right_invertible & M is left_invertible ) );
d34_algstr_0:: Trivial-multLoopStr_0 = multLoopStr_0(# 1,op2,op0,op0 #);
d35_algstr_0:: for M being multLoopStr_0 for x, b3 being Element of M holds ( ( x is left_invertible & x is right_mult-cancelable implies ( b3 = x " iff b3 * x = 1. M ) ) & ( ( not x is left_invertible or not x is right_mult-cancelable ) implies ( b3 = x " iff b3 = 0. M ) ) );
d36_algstr_0:: for M being multLoopStr_0 holds ( M is almost_left_cancelable iff for x being Element of M st x <> 0. M holds x is left_mult-cancelable );
d37_algstr_0:: for M being multLoopStr_0 holds ( M is almost_right_cancelable iff for x being Element of M st x <> 0. M holds x is right_mult-cancelable );
d38_algstr_0:: for M being multLoopStr_0 holds ( M is almost_cancelable iff ( M is almost_left_cancelable & M is almost_right_cancelable ) );
d39_algstr_0:: for M being multLoopStr_0 holds ( M is almost_left_invertible iff for x being Element of M st x <> 0. M holds x is left_invertible );
d4_algstr_0:: for M being addMagma for x being Element of M holds ( x is right_add-cancelable iff for y, z being Element of M st y + x = z + x holds y = z );
d40_algstr_0:: for M being multLoopStr_0 holds ( M is almost_right_invertible iff for x being Element of M st x <> 0. M holds x is right_invertible );
d41_algstr_0:: for M being multLoopStr_0 holds ( M is almost_invertible iff ( M is almost_right_invertible & M is almost_left_invertible ) );
d42_algstr_0:: Trivial-doubleLoopStr = doubleLoopStr(# 1,op2,op2,op0,op0 #);
d5_algstr_0:: for M being addMagma for x being Element of M holds ( x is add-cancelable iff ( x is right_add-cancelable & x is left_add-cancelable ) );
d6_algstr_0:: for M being addMagma holds ( M is left_add-cancelable iff for x being Element of M holds x is left_add-cancelable );
d7_algstr_0:: for M being addMagma holds ( M is right_add-cancelable iff for x being Element of M holds x is right_add-cancelable );
d8_algstr_0:: for M being addMagma holds ( M is add-cancelable iff ( M is right_add-cancelable & M is left_add-cancelable ) );
d9_algstr_0:: Trivial-addLoopStr = addLoopStr(# 1,op2,op0 #);
d1_algstr_1:: for x being set holds Extract x = x;
d10_algstr_1:: for IT being non empty multLoopStr_0 holds ( IT is multLoop_0-like iff ( IT is almost_invertible & IT is almost_cancelable & ( for a being Element of IT holds a * (0. IT) = 0. IT ) & ( for a being Element of IT holds (0. IT) * a = 0. IT ) ) );
d11_algstr_1:: for L being non empty almost_cancelable almost_invertible multLoopStr_0 for x being Element of L st x <> 0. L holds for b3 being Element of the carrier of L holds ( b3 = x " iff b3 * x = 1. L );
d12_algstr_1:: for L being non empty almost_cancelable almost_invertible multLoopStr_0 for a, b being Element of L holds a / b = a * (b ");
d2_algstr_1:: for IT being non empty addLoopStr holds ( IT is left_zeroed iff for a being Element of IT holds (0. IT) + a = a );
d3_algstr_1:: for L being non empty addLoopStr holds ( L is add-left-invertible iff for a, b being Element of L ex x being Element of L st x + a = b );
d4_algstr_1:: for L being non empty addLoopStr holds ( L is add-right-invertible iff for a, b being Element of L ex x being Element of L st a + x = b );
d5_algstr_1:: for IT being non empty addLoopStr holds ( IT is Loop-like iff ( IT is left_add-cancelable & IT is right_add-cancelable & IT is add-left-invertible & IT is add-right-invertible ) );
d6_algstr_1:: for IT being non empty multLoopStr holds ( IT is invertible iff ( ( for a, b being Element of IT ex x being Element of IT st a * x = b ) & ( for a, b being Element of IT ex x being Element of IT st x * a = b ) ) );
d7_algstr_1:: for L being non empty cancelable invertible multLoopStr for a, b being Element of L holds a / b = a * (b ");
d8_algstr_1:: multEX_0 = multLoopStr_0(# REAL,multreal,0,1 #);
d9_algstr_1:: for IT being non empty multLoopStr_0 holds ( IT is almost_invertible iff ( ( for a, b being Element of IT st a <> 0. IT holds ex x being Element of IT st a * x = b ) & ( for a, b being Element of IT st a <> 0. IT holds ex x being Element of IT st x * a = b ) ) );
t1_algstr_1:: for L being non empty addLoopStr for a, b being Element of L st ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) & a + b = 0. L holds b + a = 0. L
t10_algstr_1:: for a, b being Element of Trivial-multLoopStr holds a * b = 1. Trivial-multLoopStr by Th9;
t11_algstr_1:: for L being non empty multLoopStr holds ( L is multGroup iff ( ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L ex x being Element of L st a * x = 1. L ) & ( for a, b, c being Element of L holds (a * b) * c = a * (b * c) ) ) )
t12_algstr_1:: for L being non empty multLoopStr holds ( L is commutative multGroup iff ( ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L ex x being Element of L st a * x = 1. L ) & ( for a, b, c being Element of L holds (a * b) * c = a * (b * c) ) & ( for a, b being Element of L holds a * b = b * a ) ) ) by Th11, GROUP_1:def_12;
t13_algstr_1:: for G being multGroup for a being Element of G holds ( (a ") * a = 1. G & a * (a ") = 1. G )
t14_algstr_1:: for q, p being Real st q <> 0 holds ex y being Real st p = q * y
t15_algstr_1:: for q, p being Real st q <> 0 holds ex y being Real st p = y * q
t16_algstr_1:: for L being non empty multLoopStr_0 holds ( L is multLoop_0-like iff ( ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st a * x = b ) & ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st x * a = b ) & ( for a, x, y being Element of L st a <> 0. L & a * x = a * y holds x = y ) & ( for a, x, y being Element of L st a <> 0. L & x * a = y * a holds x = y ) & ( for a being Element of L holds a * (0. L) = 0. L ) & ( for a being Element of L holds (0. L) * a = 0. L ) ) )
t17_algstr_1:: for G being non empty almost_cancelable associative well-unital almost_invertible multLoopStr_0 for a being Element of G st a <> 0. G holds ( (a ") * a = 1. G & a * (a ") = 1. G )
t2_algstr_1:: for L being non empty addLoopStr for a being Element of L st ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) holds (0. L) + a = a + (0. L)
t3_algstr_1:: for L being non empty addLoopStr st ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) holds for a being Element of L ex x being Element of L st x + a = 0. L
t4_algstr_1:: for a, b being Element of Trivial-addLoopStr holds a = b
t5_algstr_1:: for a, b being Element of Trivial-addLoopStr holds a + b = 0. Trivial-addLoopStr by Th4;
t6_algstr_1:: for L being non empty addLoopStr holds ( L is Loop-like iff ( ( for a, b being Element of L ex x being Element of L st a + x = b ) & ( for a, b being Element of L ex x being Element of L st x + a = b ) & ( for a, x, y being Element of L st a + x = a + y holds x = y ) & ( for a, x, y being Element of L st x + a = y + a holds x = y ) ) )
t7_algstr_1:: for L being non empty addLoopStr holds ( L is AddGroup iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) ) )
t8_algstr_1:: for L being non empty addLoopStr holds ( L is Abelian AddGroup iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) & ( for a, b being Element of L holds a + b = b + a ) ) ) by Th7, RLVECT_1:def_2;
t9_algstr_1:: for a, b being Element of Trivial-multLoopStr holds a = b
d1_algstr_2:: for L being non empty left_add-cancelable add-right-invertible addLoopStr for a, b3 being Element of L holds ( b3 = - a iff a + b3 = 0. L );
d2_algstr_2:: for L being non empty left_add-cancelable add-right-invertible addLoopStr for a, b being Element of L holds a - b = a + (- b);
t1_algstr_2:: for L being non empty doubleLoopStr holds ( L is leftQuasi-Field iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) & ( for a, b being Element of L holds a + b = b + a ) & 0. L <> 1. L & ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L holds (1. L) * a = a ) & ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st a * x = b ) & ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st x * a = b ) & ( for a, x, y being Element of L st a <> 0. L & a * x = a * y holds x = y ) & ( for a, x, y being Element of L st a <> 0. L & x * a = y * a holds x = y ) & ( for a being Element of L holds a * (0. L) = 0. L ) & ( for a being Element of L holds (0. L) * a = 0. L ) & ( for a, b, c being Element of L holds a * (b + c) = (a * b) + (a * c) ) ) )
t10_algstr_2:: for L being non empty doubleLoopStr holds ( L is doublesidedQuasi-Field iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) & ( for a, b being Element of L holds a + b = b + a ) & 0. L <> 1. L & ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L holds (1. L) * a = a ) & ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st a * x = b ) & ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st x * a = b ) & ( for a, x, y being Element of L st a <> 0. L & a * x = a * y holds x = y ) & ( for a, x, y being Element of L st a <> 0. L & x * a = y * a holds x = y ) & ( for a being Element of L holds a * (0. L) = 0. L ) & ( for a being Element of L holds (0. L) * a = 0. L ) & ( for a, b, c being Element of L holds a * (b + c) = (a * b) + (a * c) ) & ( for a, b, c being Element of L holds (b + c) * a = (b * a) + (c * a) ) ) )
t11_algstr_2:: for L being non empty doubleLoopStr holds ( L is _Skew-Field iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) & ( for a, b being Element of L holds a + b = b + a ) & 0. L <> 1. L & ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L st a <> 0. L holds ex x being Element of L st a * x = 1. L ) & ( for a being Element of L holds a * (0. L) = 0. L ) & ( for a being Element of L holds (0. L) * a = 0. L ) & ( for a, b, c being Element of L holds (a * b) * c = a * (b * c) ) & ( for a, b, c being Element of L holds a * (b + c) = (a * b) + (a * c) ) & ( for a, b, c being Element of L holds (b + c) * a = (b * a) + (c * a) ) ) )
t12_algstr_2:: for L being non empty doubleLoopStr holds ( L is _Field iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) & ( for a, b being Element of L holds a + b = b + a ) & 0. L <> 1. L & ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L st a <> 0. L holds ex x being Element of L st a * x = 1. L ) & ( for a being Element of L holds a * (0. L) = 0. L ) & ( for a, b, c being Element of L holds (a * b) * c = a * (b * c) ) & ( for a, b, c being Element of L holds a * (b + c) = (a * b) + (a * c) ) & ( for a, b being Element of L holds a * b = b * a ) ) )
t2_algstr_2:: for G being Abelian right-distributive doubleLoop for a, b being Element of G holds a * (- b) = - (a * b)
t3_algstr_2:: for G being non empty left_add-cancelable add-right-invertible Abelian addLoopStr for a being Element of G holds - (- a) = a
t4_algstr_2:: for G being Abelian right-distributive doubleLoop holds (- (1. G)) * (- (1. G)) = 1. G
t5_algstr_2:: for G being Abelian right-distributive doubleLoop for a, x, y being Element of G holds a * (x - y) = (a * x) - (a * y)
t6_algstr_2:: for L being non empty doubleLoopStr holds ( L is rightQuasi-Field iff ( ( for a being Element of L holds a + (0. L) = a ) & ( for a being Element of L ex x being Element of L st a + x = 0. L ) & ( for a, b, c being Element of L holds (a + b) + c = a + (b + c) ) & ( for a, b being Element of L holds a + b = b + a ) & 0. L <> 1. L & ( for a being Element of L holds a * (1. L) = a ) & ( for a being Element of L holds (1. L) * a = a ) & ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st a * x = b ) & ( for a, b being Element of L st a <> 0. L holds ex x being Element of L st x * a = b ) & ( for a, x, y being Element of L st a <> 0. L & a * x = a * y holds x = y ) & ( for a, x, y being Element of L st a <> 0. L & x * a = y * a holds x = y ) & ( for a being Element of L holds a * (0. L) = 0. L ) & ( for a being Element of L holds (0. L) * a = 0. L ) & ( for a, b, c being Element of L holds (b + c) * a = (b * a) + (c * a) ) ) )
t7_algstr_2:: for G being left-distributive doubleLoop for b, a being Element of G holds (- b) * a = - (b * a)
t8_algstr_2:: for G being Abelian left-distributive doubleLoop holds (- (1. G)) * (- (1. G)) = 1. G
t9_algstr_2:: for G being left-distributive doubleLoop for x, y, a being Element of G holds (x - y) * a = (x * a) - (y * a)
d1_algstr_3:: for F being non empty TernaryFieldStr for a, b, c being Scalar of F holds Tern (a,b,c) = the TernOp of F . (a,b,c);
d2_algstr_3:: for b1 being TriOp of REAL holds ( b1 = ternaryreal iff for a, b, c being Real holds b1 . (a,b,c) = (a * b) + c );
d3_algstr_3:: TernaryFieldEx = TernaryFieldStr(# REAL,0,1,ternaryreal #);
d4_algstr_3:: for a, b, c being Scalar of TernaryFieldEx holds tern (a,b,c) = the TernOp of TernaryFieldEx . (a,b,c);
d5_algstr_3:: for IT being non empty TernaryFieldStr holds ( IT is Ternary-Field-like iff ( 0. IT <> 1. IT & ( for a being Scalar of IT holds Tern (a,(1. IT),(0. IT)) = a ) & ( for a being Scalar of IT holds Tern ((1. IT),a,(0. IT)) = a ) & ( for a, b being Scalar of IT holds Tern (a,(0. IT),b) = b ) & ( for a, b being Scalar of IT holds Tern ((0. IT),a,b) = b ) & ( for u, a, b being Scalar of IT ex v being Scalar of IT st Tern (u,a,v) = b ) & ( for u, a, v, v9 being Scalar of IT st Tern (u,a,v) = Tern (u,a,v9) holds v = v9 ) & ( for a, a9 being Scalar of IT st a <> a9 holds for b, b9 being Scalar of IT ex u, v being Scalar of IT st ( Tern (u,a,v) = b & Tern (u,a9,v) = b9 ) ) & ( for u, u9 being Scalar of IT st u <> u9 holds for v, v9 being Scalar of IT ex a being Scalar of IT st Tern (u,a,v) = Tern (u9,a,v9) ) & ( for a, a9, u, u9, v, v9 being Scalar of IT st Tern (u,a,v) = Tern (u9,a,v9) & Tern (u,a9,v) = Tern (u9,a9,v9) & not a = a9 holds u = u9 ) ) );
t1_algstr_3:: for u, u9, v, v9 being Real st u <> u9 holds ex x being Real st (u * x) + v = (u9 * x) + v9
t2_algstr_3:: for u, a, v being Scalar of TernaryFieldEx for z, x, y being Real st u = z & a = x & v = y holds Tern (u,a,v) = (z * x) + y by Def2;
t3_algstr_3:: 1 = 1. TernaryFieldEx ;
t4_algstr_3:: for F being Ternary-Field for a, a9, u, v, u9, v9 being Scalar of F st a <> a9 & Tern (u,a,v) = Tern (u9,a,v9) & Tern (u,a9,v) = Tern (u9,a9,v9) holds ( u = u9 & v = v9 )
t5_algstr_3:: for F being Ternary-Field for a being Scalar of F st a <> 0. F holds for b, c being Scalar of F ex x being Scalar of F st Tern (a,x,b) = c
t6_algstr_3:: for F being Ternary-Field for a, x, b, x9 being Scalar of F st a <> 0. F & Tern (a,x,b) = Tern (a,x9,b) holds x = x9
t7_algstr_3:: for F being Ternary-Field for a being Scalar of F st a <> 0. F holds for b, c being Scalar of F ex x being Scalar of F st Tern (x,a,b) = c
t8_algstr_3:: for F being Ternary-Field for a, x, b, x9 being Scalar of F st a <> 0. F & Tern (x,a,b) = Tern (x9,a,b) holds x = x9
d1_algstr_4:: for X, x being set holds ( ( x is XFinSequence of implies IFXFinSequence (x,X) = x ) & ( x is not XFinSequence of implies IFXFinSequence (x,X) = <%> X ) );
d10_algstr_4:: for M being multMagma for A being Subset of M holds ( A is stable iff for v, w being Element of M st v in A & w in A holds v * w in A );
d11_algstr_4:: for M being multMagma for A being stable Subset of M holds the_mult_induced_by A = the multF of M || A;
d12_algstr_4:: for M being multMagma for A being Subset of M for b3 being strict multSubmagma of M holds ( b3 = the_submagma_generated_by A iff ( A c= the carrier of b3 & ( for N being strict multSubmagma of M st A c= the carrier of N holds b3 is multSubmagma of N ) ) );
d13_algstr_4:: for X being set for b2 being Function of NAT,(bool (the_universe_of (X \/ NAT))) holds ( b2 = free_magma_seq X iff ( b2 . 0 = {} & b2 . 1 = X & ( for n being Nat st n >= 2 holds ex fs being FinSequence st ( len fs = n - 1 & ( for p being Nat st p >= 1 & p <= n - 1 holds fs . p = [:(b2 . p),(b2 . (n - p)):] ) & b2 . n = Union (disjoin fs) ) ) ) );
d14_algstr_4:: for X being set for n being Nat holds free_magma (X,n) = (free_magma_seq X) . n;
d15_algstr_4:: for X being set holds free_magma_carrier X = Union (disjoin ((free_magma_seq X) | NATPLUS));
d16_algstr_4:: for X being set for b2 being BinOp of (free_magma_carrier X) holds ( ( not X is empty implies ( b2 = free_magma_mult X iff for v, w being Element of free_magma_carrier X for n, m being Nat st n = v `2 & m = w `2 holds b2 . (v,w) = [[[(v `1),(w `1)],(v `2)],(n + m)] ) ) & ( X is empty implies ( b2 = free_magma_mult X iff b2 = {} ) ) );
d17_algstr_4:: for X being set holds free_magma X = multMagma(# (free_magma_carrier X),(free_magma_mult X) #);
d18_algstr_4:: for X being set for w being Element of (free_magma X) holds ( ( not X is empty implies length w = w `2 ) & ( X is empty implies length w = 0 ) );
d19_algstr_4:: for X being set for n being Nat for b3 being Function of (free_magma (X,n)),(free_magma X) holds ( ( n > 0 implies ( b3 = canon_image (X,n) iff for x being set st x in dom b3 holds b3 . x = [x,n] ) ) & ( not n > 0 implies ( b3 = canon_image (X,n) iff b3 = {} ) ) );
d2_algstr_4:: for f, g being Function holds ( f extends g iff ( dom g c= dom f & f tolerates g ) );
d20_algstr_4:: for X being non empty set for M being non empty multMagma for n, m being non zero Nat for f being Function of (free_magma (X,n)),M for g being Function of (free_magma (X,m)),M for b7 being Function of [:[:(free_magma (X,n)),(free_magma (X,m)):],{n}:],M holds ( b7 = [:f,g:] iff for x being Element of [:[:(free_magma (X,n)),(free_magma (X,m)):],{n}:] for y being Element of free_magma (X,n) for z being Element of free_magma (X,m) st y = (x `1) `1 & z = (x `1) `2 holds b7 . x = (f . y) * (g . z) );
d21_algstr_4:: for X, Y being non empty set for f being Function of X,Y for b4 being Function of (free_magma X),(free_magma Y) holds ( b4 = free_magmaF f iff ( b4 is multiplicative & b4 extends ((canon_image (Y,1)) * f) * ((canon_image (X,1)) ") ) );
d3_algstr_4:: for M being multMagma for R being Equivalence_Relation of M holds ( R is compatible iff for v, v9, w, w9 being Element of M st v in Class (R,v9) & w in Class (R,w9) holds v * w in Class (R,(v9 * w9)) );
d4_algstr_4:: for M being multMagma for R being compatible Equivalence_Relation of M for b3 being BinOp of (Class R) holds ( ( not M is empty implies ( b3 = ClassOp R iff for x, y being Element of Class R for v, w being Element of M st x = Class (R,v) & y = Class (R,w) holds b3 . (x,y) = Class (R,(v * w)) ) ) & ( M is empty implies ( b3 = ClassOp R iff b3 = {} ) ) );
d5_algstr_4:: for M being multMagma for R being compatible Equivalence_Relation of M holds M ./. R = multMagma(# (Class R),(ClassOp R) #);
d6_algstr_4:: for M being non empty multMagma for R being compatible Equivalence_Relation of M for b3 being Function of M,(M ./. R) holds ( b3 = nat_hom R iff for v being Element of M holds b3 . v = Class (R,v) );
d7_algstr_4:: for M being multMagma for r being Relators of M holds equ_rel r = meet { R where R is compatible Equivalence_Relation of M : for i being set for v, w being Element of M st i in dom r & r . i = [v,w] holds v in Class (R,w) } ;
d8_algstr_4:: for X, Y being set for f being Function of X,Y for b4 being Equivalence_Relation of X holds ( b4 = equ_kernel f iff for x, y being set holds ( [x,y] in b4 iff ( x in X & y in X & f . x = f . y ) ) );
d9_algstr_4:: for M, b2 being multMagma holds ( b2 is multSubmagma of M iff ( the carrier of b2 c= the carrier of M & the multF of b2 = the multF of M || the carrier of b2 ) );
s1_algstr_4:: scheme FuncRecursiveUniq{ F1( set ) -> set , F2() -> Function, F3() -> Function } : F2() = F3() provided A1: ( dom F2() = NAT & ( for n being Nat holds F2() . n = F1((F2() | n)) ) ) and A2: ( dom F3() = NAT & ( for n being Nat holds F3() . n = F1((F3() | n)) ) )
s2_algstr_4:: scheme FuncRecursiveExist{ F1( set ) -> set } : ex f being Function st ( dom f = NAT & ( for n being Nat holds f . n = F1((f | n)) ) )
s3_algstr_4:: scheme FuncRecursiveUniqu2{ F1() -> non empty set , F2( XFinSequence of ) -> Element of F1(), F3() -> Function of NAT,F1(), F4() -> Function of NAT,F1() } : F3() = F4() provided A1: for n being Nat holds F3() . n = F2((F3() | n)) and A2: for n being Nat holds F4() . n = F2((F4() | n))
s4_algstr_4:: scheme FuncRecursiveExist2{ F1() -> non empty set , F2( XFinSequence of ) -> Element of F1() } : ex f being Function of NAT,F1() st for n being Nat holds f . n = F2((f | n))
t1_algstr_4:: for X, Y, Z being set st Y c= the_universe_of X & Z c= the_universe_of X holds [:Y,Z:] c= the_universe_of X
t10_algstr_4:: for M being non empty multMagma for A being Subset of M holds ( A is stable iff A * A c= A )
t11_algstr_4:: for M, N being non empty multMagma for f being Function of M,N for X being stable Subset of M st f is multiplicative holds f .: X is stable Subset of N
t12_algstr_4:: for N, M being non empty multMagma for f being Function of M,N for Y being stable Subset of N st f is multiplicative holds f " Y is stable Subset of M
t13_algstr_4:: for N, M being non empty multMagma for f, g being Function of M,N st f is multiplicative & g is multiplicative holds { v where v is Element of M : f . v = g . v } is stable Subset of M
t14_algstr_4:: for M being multMagma for A being Subset of M holds ( A is empty iff the_submagma_generated_by A is empty )
t15_algstr_4:: for M, N being non empty multMagma for f being Function of M,N for X being Subset of M st f is multiplicative holds f .: the carrier of (the_submagma_generated_by X) = the carrier of (the_submagma_generated_by (f .: X))
t16_algstr_4:: for X being set holds free_magma (X,0) = {} by Def13;
t17_algstr_4:: for X being set holds free_magma (X,1) = X by Def13;
t18_algstr_4:: for X being set holds free_magma (X,2) = [:[:X,X:],{1}:]
t19_algstr_4:: for X being set holds free_magma (X,3) = [:[:X,[:[:X,X:],{1}:]:],{1}:] \/ [:[:[:[:X,X:],{1}:],X:],{2}:]
t2_algstr_4:: for M being multMagma for R being Equivalence_Relation of M holds ( R is compatible iff for v, v9, w, w9 being Element of M st Class (R,v) = Class (R,v9) & Class (R,w) = Class (R,w9) holds Class (R,(v * w)) = Class (R,(v9 * w9)) )
t20_algstr_4:: for X being set for n being Nat st n >= 2 holds ex fs being FinSequence st ( len fs = n - 1 & ( for p being Nat st p >= 1 & p <= n - 1 holds fs . p = [:(free_magma (X,p)),(free_magma (X,(n -' p))):] ) & free_magma (X,n) = Union (disjoin fs) )
t21_algstr_4:: for X, x being set for n being Nat st n >= 2 & x in free_magma (X,n) holds ex p, m being Nat st ( x `2 = p & 1 <= p & p <= n - 1 & (x `1) `1 in free_magma (X,p) & (x `1) `2 in free_magma (X,m) & n = p + m & x in [:[:(free_magma (X,p)),(free_magma (X,m)):],{p}:] )
t22_algstr_4:: for X, x, y being set for n, m being Nat st x in free_magma (X,n) & y in free_magma (X,m) holds [[x,y],n] in free_magma (X,(n + m))
t23_algstr_4:: for X, Y being set for n being Nat st X c= Y holds free_magma (X,n) c= free_magma (Y,n)
t24_algstr_4:: for X being set holds ( X = {} iff free_magma_carrier X = {} )
t25_algstr_4:: for X being non empty set for w being Element of free_magma_carrier X holds w in [:(free_magma (X,(w `2))),{(w `2)}:]
t26_algstr_4:: for X being non empty set for v, w being Element of free_magma_carrier X holds [[[(v `1),(w `1)],(v `2)],((v `2) + (w `2))] is Element of free_magma_carrier X
t27_algstr_4:: for X, Y being set st X c= Y holds free_magma_carrier X c= free_magma_carrier Y
t28_algstr_4:: for X being set for n being Nat st n > 0 holds [:(free_magma (X,n)),{n}:] c= free_magma_carrier X by Lm1;
t29_algstr_4:: for X being set for S being Subset of X holds free_magma S is multSubmagma of free_magma X
t3_algstr_4:: for M being multMagma for r being Relators of M for R being compatible Equivalence_Relation of M st ( for i being set for v, w being Element of M st i in dom r & r . i = [v,w] holds v in Class (R,w) ) holds equ_rel r c= R
t30_algstr_4:: for X being set holds X = { (w `1) where w is Element of (free_magma X) : length w = 1 }
t31_algstr_4:: for X being set for v, w being Element of (free_magma X) st not X is empty holds v * w = [[[(v `1),(w `1)],(v `2)],((length v) + (length w))]
t32_algstr_4:: for X being set for v being Element of (free_magma X) st not X is empty holds ( v = [(v `1),(v `2)] & length v >= 1 )
t33_algstr_4:: for X being set for v, w being Element of (free_magma X) holds length (v * w) = (length v) + (length w)
t34_algstr_4:: for X being set for w being Element of (free_magma X) st length w >= 2 holds ex w1, w2 being Element of (free_magma X) st ( w = w1 * w2 & length w1 < length w & length w2 < length w )
t35_algstr_4:: for X being set for v1, v2, w1, w2 being Element of (free_magma X) st v1 * v2 = w1 * w2 holds ( v1 = w1 & v2 = w2 )
t36_algstr_4:: for X being non empty set for A being Subset of (free_magma X) st A = (canon_image (X,1)) .: X holds free_magma X = the_submagma_generated_by A
t37_algstr_4:: for X being non empty set for R being compatible Equivalence_Relation of (free_magma X) holds (free_magma X) ./. R = the_submagma_generated_by ((nat_hom R) .: ((canon_image (X,1)) .: X))
t38_algstr_4:: for X, Y being non empty set for f being Function of X,Y holds (canon_image (Y,1)) * f is Function of X,(free_magma Y)
t39_algstr_4:: for X being non empty set for M being non empty multMagma for f being Function of X,M ex h being Function of (free_magma X),M st ( h is multiplicative & h extends f * ((canon_image (X,1)) ") )
t4_algstr_4:: for M, N being non empty multMagma for f being Function of M,N st f is multiplicative holds equ_kernel f is compatible
t40_algstr_4:: for X being non empty set for M being non empty multMagma for f being Function of X,M for h, g being Function of (free_magma X),M st h is multiplicative & h extends f * ((canon_image (X,1)) ") & g is multiplicative & g extends f * ((canon_image (X,1)) ") holds h = g
t41_algstr_4:: for N, M being non empty multMagma for f being Function of M,N for H being non empty multSubmagma of N for R being compatible Equivalence_Relation of M st f is multiplicative & the carrier of H = rng f & R = equ_kernel f holds ex g being Function of (M ./. R),H st ( f = g * (nat_hom R) & g is bijective & g is multiplicative )
t42_algstr_4:: for M, N being non empty multMagma for R being compatible Equivalence_Relation of M for g1, g2 being Function of (M ./. R),N st g1 * (nat_hom R) = g2 * (nat_hom R) holds g1 = g2
t43_algstr_4:: for M being non empty multMagma ex X being non empty set ex r being Relators of (free_magma X) ex g being Function of ((free_magma X) ./. (equ_rel r)),M st ( g is bijective & g is multiplicative )
t44_algstr_4:: for X, Y, Z being non empty set for f being Function of X,Y for g being Function of Y,Z holds free_magmaF (g * f) = (free_magmaF g) * (free_magmaF f)
t45_algstr_4:: for X, Z, Y being non empty set for f being Function of X,Y for g being Function of X,Z st Y c= Z & f = g holds free_magmaF f = free_magmaF g
t46_algstr_4:: for Y, X being non empty set for f being Function of X,Y holds free_magmaF (id X) = id (dom (free_magmaF f))
t47_algstr_4:: for X, Y being non empty set for f being Function of X,Y st f is one-to-one holds free_magmaF f is one-to-one
t48_algstr_4:: for X, Y being non empty set for f being Function of X,Y st f is onto holds free_magmaF f is onto
t5_algstr_4:: for N, M being non empty multMagma for f being Function of M,N st f is multiplicative holds ex r being Relators of M st equ_kernel f = equ_rel r
t6_algstr_4:: for M being multMagma for N, K being multSubmagma of M st N is multSubmagma of K & K is multSubmagma of N holds multMagma(# the carrier of N, the multF of N #) = multMagma(# the carrier of K, the multF of K #)
t7_algstr_4:: for M being multMagma for N being multSubmagma of M st the carrier of N = the carrier of M holds multMagma(# the carrier of N, the multF of N #) = multMagma(# the carrier of M, the multF of M #)
t8_algstr_4:: for M being multMagma for N being multSubmagma of M holds the carrier of N is stable Subset of M
t9_algstr_4:: for M being multMagma for F being Function st ( for i being set st i in dom F holds F . i is stable Subset of M ) holds meet F is stable Subset of M
d1_ali2:: for M being non empty MetrSpace for f being Function of M,M holds ( f is contraction iff ex L being Real st ( 0 < L & L < 1 & ( for x, y being Point of M holds dist ((f . x),(f . y)) <= L * (dist (x,y)) ) ) );
t1_ali2:: for M being non empty MetrSpace for f being Contraction of M st TopSpaceMetr M is compact holds ex c being Point of M st ( f . c = c & ( for x being Point of M st f . x = x holds x = c ) )
d1_altcat_1:: for G being AltGraph for o1, o2 being object of G holds <^o1,o2^> = the Arrows of G . (o1,o2);
d10_altcat_1:: for A, B being functional set for b3 being compositional ManySortedFunction of [:B,A:] holds ( b3 = FuncComp (A,B) iff verum );
d11_altcat_1:: for C being non empty AltCatStr holds ( C is quasi-functional iff for a1, a2 being object of C holds <^a1,a2^> c= Funcs (a1,a2) );
d12_altcat_1:: for C being non empty AltCatStr holds ( C is semi-functional iff for a1, a2, a3 being object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} holds for f being Morphism of a1,a2 for g being Morphism of a2,a3 for f9, g9 being Function st f = f9 & g = g9 holds g * f = g9 * f9 );
d13_altcat_1:: for C being non empty AltCatStr holds ( C is pseudo-functional iff for o1, o2, o3 being object of C holds the Comp of C . (o1,o2,o3) = (FuncComp ((Funcs (o1,o2)),(Funcs (o2,o3)))) | [:<^o2,o3^>,<^o1,o2^>:] );
d14_altcat_1:: for A being non empty set for b2 being non empty strict pseudo-functional AltCatStr holds ( b2 = EnsCat A iff ( the carrier of b2 = A & ( for a1, a2 being object of b2 holds <^a1,a2^> = Funcs (a1,a2) ) ) );
d15_altcat_1:: for C being non empty AltCatStr holds ( C is associative iff the Comp of C is associative );
d16_altcat_1:: for C being non empty AltCatStr holds ( C is with_units iff ( the Comp of C is with_left_units & the Comp of C is with_right_units ) );
d17_altcat_1:: for C being non empty with_units AltCatStr for o being object of C for b3 being Morphism of o,o holds ( b3 = idm o iff for o9 being object of C st <^o,o9^> <> {} holds for a being Morphism of o,o9 holds a * b3 = a );
d18_altcat_1:: for C being AltCatStr holds ( C is quasi-discrete iff for i, j being object of C st <^i,j^> <> {} holds i = j );
d19_altcat_1:: for C being AltCatStr holds ( C is pseudo-discrete iff for i being object of C holds <^i,i^> is trivial );
d2_altcat_1:: for G being AltGraph holds ( G is transitive iff for o1, o2, o3 being object of G st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds <^o1,o3^> <> {} );
d20_altcat_1:: for A being non empty set for b2 being non empty strict quasi-discrete AltCatStr holds ( b2 = DiscrCat A iff ( the carrier of b2 = A & ( for i being object of b2 holds <^i,i^> = {(id i)} ) ) );
d3_altcat_1:: for I being set for G being ManySortedSet of [:I,I:] for b3 being ManySortedSet of [:I,I,I:] holds ( b3 = {|G|} iff for i, j, k being set st i in I & j in I & k in I holds b3 . (i,j,k) = G . (i,k) );
d4_altcat_1:: for I being set for G, H being ManySortedSet of [:I,I:] for b4 being ManySortedSet of [:I,I,I:] holds ( b4 = {|G,H|} iff for i, j, k being set st i in I & j in I & k in I holds b4 . (i,j,k) = [:(H . (j,k)),(G . (i,j)):] );
d5_altcat_1:: for I being non empty set for G being ManySortedSet of [:I,I:] for IT being BinComp of G holds ( IT is associative iff for i, j, k, l being Element of I for f, g, h being set st f in G . (i,j) & g in G . (j,k) & h in G . (k,l) holds (IT . (i,k,l)) . (h,((IT . (i,j,k)) . (g,f))) = (IT . (i,j,l)) . (((IT . (j,k,l)) . (h,g)),f) );
d6_altcat_1:: for I being non empty set for G being ManySortedSet of [:I,I:] for IT being BinComp of G holds ( IT is with_right_units iff for i being Element of I ex e being set st ( e in G . (i,i) & ( for j being Element of I for f being set st f in G . (i,j) holds (IT . (i,i,j)) . (f,e) = f ) ) );
d7_altcat_1:: for I being non empty set for G being ManySortedSet of [:I,I:] for IT being BinComp of G holds ( IT is with_left_units iff for j being Element of I ex e being set st ( e in G . (j,j) & ( for i being Element of I for f being set st f in G . (i,j) holds (IT . (i,j,j)) . (e,f) = f ) ) );
d8_altcat_1:: for C being non empty AltCatStr for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds for f being Morphism of o1,o2 for g being Morphism of o2,o3 holds g * f = ( the Comp of C . (o1,o2,o3)) . (g,f);
d9_altcat_1:: for IT being Function holds ( IT is compositional iff for x being set st x in dom IT holds ex f, g being Function st ( x = [g,f] & IT . x = g * f ) );
s1_altcat_1:: scheme MSSLambda2{ F1() -> set , F2() -> set , F3( set , set ) -> set } : ex M being ManySortedSet of [:F1(),F2():] st for i, j being set st i in F1() & j in F2() holds M . (i,j) = F3(i,j)
s2_altcat_1:: scheme MSSLambda2D{ F1() -> non empty set , F2() -> non empty set , F3( set , set ) -> set } : ex M being ManySortedSet of [:F1(),F2():] st for i being Element of F1() for j being Element of F2() holds M . (i,j) = F3(i,j)
s3_altcat_1:: scheme MSSLambda3{ F1() -> set , F2() -> set , F3() -> set , F4( set , set , set ) -> set } : ex M being ManySortedSet of [:F1(),F2(),F3():] st for i, j, k being set st i in F1() & j in F2() & k in F3() holds M . (i,j,k) = F4(i,j,k)
s4_altcat_1:: scheme MSSLambda3D{ F1() -> non empty set , F2() -> non empty set , F3() -> non empty set , F4( set , set , set ) -> set } : ex M being ManySortedSet of [:F1(),F2(),F3():] st for i being Element of F1() for j being Element of F2() for k being Element of F3() holds M . (i,j,k) = F4(i,j,k)
t1_altcat_1:: for A being set holds id A in Funcs (A,A)
t10_altcat_1:: for i, j, k being set holds ((i,j) :-> k) . (i,j) = k
t11_altcat_1:: for A, B being functional set for F being compositional ManySortedSet of [:A,B:] for g, f being Function st g in A & f in B holds F . (g,f) = g * f
t12_altcat_1:: for A, B, C being set holds rng (FuncComp ((Funcs (A,B)),(Funcs (B,C)))) c= Funcs (A,C)
t13_altcat_1:: for o being set holds FuncComp ({(id o)},{(id o)}) = ((id o),(id o)) :-> (id o)
t14_altcat_1:: for A, B being functional set for A1 being Subset of A for B1 being Subset of B holds FuncComp (A1,B1) = (FuncComp (A,B)) | [:B1,A1:]
t15_altcat_1:: for C being non empty AltCatStr for a1, a2, a3 being object of C holds the Comp of C . (a1,a2,a3) is Function of [:<^a2,a3^>,<^a1,a2^>:],<^a1,a3^> ;
t16_altcat_1:: for C being non empty pseudo-functional AltCatStr for a1, a2, a3 being object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} holds for f being Morphism of a1,a2 for g being Morphism of a2,a3 for f9, g9 being Function st f = f9 & g = g9 holds g * f = g9 * f9
t17_altcat_1:: for C being non empty transitive AltCatStr for a1, a2, a3 being object of C holds ( dom ( the Comp of C . (a1,a2,a3)) = [:<^a2,a3^>,<^a1,a2^>:] & rng ( the Comp of C . (a1,a2,a3)) c= <^a1,a3^> )
t18_altcat_1:: for C being non empty with_units AltCatStr for o being object of C holds <^o,o^> <> {}
t19_altcat_1:: for C being non empty with_units AltCatStr for o being object of C holds idm o in <^o,o^>
t2_altcat_1:: Funcs ({},{}) = {(id {})}
t20_altcat_1:: for C being non empty with_units AltCatStr for o1, o2 being object of C st <^o1,o2^> <> {} holds for a being Morphism of o1,o2 holds (idm o2) * a = a
t21_altcat_1:: for C being non empty transitive associative AltCatStr for o1, o2, o3, o4 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o4^> <> {} holds for a being Morphism of o1,o2 for b being Morphism of o2,o3 for c being Morphism of o3,o4 holds c * (b * a) = (c * b) * a
t22_altcat_1:: for C being non empty with_units AltCatStr holds ( C is pseudo-discrete iff for o being object of C holds <^o,o^> = {(idm o)} )
t23_altcat_1:: ( EnsCat 1 is pseudo-discrete & EnsCat 1 is 1 -element )
t24_altcat_1:: for A being non empty set for o1, o2, o3 being object of (DiscrCat A) st ( o1 <> o2 or o2 <> o3 ) holds the Comp of (DiscrCat A) . (o1,o2,o3) = {}
t25_altcat_1:: for A being non empty set for o being object of (DiscrCat A) holds the Comp of (DiscrCat A) . (o,o,o) = ((id o),(id o)) :-> (id o)
t3_altcat_1:: for A, B, C being set for f, g being Function st f in Funcs (A,B) & g in Funcs (B,C) holds g * f in Funcs (A,C)
t4_altcat_1:: for A, B, C being set st Funcs (A,B) <> {} & Funcs (B,C) <> {} holds Funcs (A,C) <> {}
t5_altcat_1:: for A, B being set for F being ManySortedSet of [:B,A:] for C being Subset of A for D being Subset of B for x, y being set st x in C & y in D holds F . (y,x) = (F | [:D,C:]) . (y,x)
t6_altcat_1:: for A, B being set for N, M being ManySortedSet of [:A,B:] st ( for i, j being set st i in A & j in B holds N . (i,j) = M . (i,j) ) holds M = N
t7_altcat_1:: for A, B being non empty set for N, M being ManySortedSet of [:A,B:] st ( for i being Element of A for j being Element of B holds N . (i,j) = M . (i,j) ) holds M = N
t8_altcat_1:: for A being set for N, M being ManySortedSet of [:A,A,A:] st ( for i, j, k being set st i in A & j in A & k in A holds N . (i,j,k) = M . (i,j,k) ) holds M = N
t9_altcat_1:: for i, j, k being set holds (i,j) :-> k = [i,j] .--> k ;
d1_altcat_2:: for f, g being Function holds ( f cc= g iff ( dom f c= dom g & ( for i being set st i in dom f holds f . i c= g . i ) ) );
d10_altcat_2:: for b1 being strict AltCatStr holds ( b1 = the_empty_category iff the carrier of b1 is empty );
d11_altcat_2:: for C, b2 being AltCatStr holds ( b2 is SubCatStr of C iff ( the carrier of b2 c= the carrier of C & the Arrows of b2 cc= the Arrows of C & the Comp of b2 cc= the Comp of C ) );
d12_altcat_2:: for C being non empty AltCatStr for o being object of C for b3 being strict SubCatStr of C holds ( b3 = ObCat o iff ( the carrier of b3 = {o} & the Arrows of b3 = (o,o) :-> <^o,o^> & the Comp of b3 = [o,o,o] .--> ( the Comp of C . (o,o,o)) ) );
d13_altcat_2:: for C being AltCatStr for D being SubCatStr of C holds ( D is full iff the Arrows of D = the Arrows of C || the carrier of D );
d14_altcat_2:: for C being non empty with_units AltCatStr for D being SubCatStr of C holds ( ( not D is empty implies ( D is id-inheriting iff for o being object of D for o9 being object of C st o = o9 holds idm o9 in <^o,o^> ) ) & ( D is empty implies ( D is id-inheriting iff verum ) ) );
d2_altcat_2:: for I, J being set for A being ManySortedSet of I for B being ManySortedSet of J holds ( A cc= B iff ( I c= J & ( for i being set st i in I holds A . i c= B . i ) ) );
d3_altcat_2:: for C being non empty non void CatStr for b2 being ManySortedSet of [: the carrier of C, the carrier of C:] holds ( b2 = the_hom_sets_of C iff for i, j being Object of C holds b2 . (i,j) = Hom (i,j) );
d4_altcat_2:: for C being Category for b2 being BinComp of (the_hom_sets_of C) holds ( b2 = the_comps_of C iff for i, j, k being Object of C holds b2 . (i,j,k) = the Comp of C | [:((the_hom_sets_of C) . (j,k)),((the_hom_sets_of C) . (i,j)):] );
d5_altcat_2:: for C being Category holds Alter C = AltCatStr(# the carrier of C,(the_hom_sets_of C),(the_comps_of C) #);
d6_altcat_2:: for C being AltGraph holds ( C is reflexive iff for x being set st x in the carrier of C holds the Arrows of C . (x,x) <> {} );
d7_altcat_2:: for C being non empty AltGraph holds ( C is reflexive iff for o being object of C holds <^o,o^> <> {} );
d8_altcat_2:: for C being non empty transitive AltCatStr holds ( C is associative iff for o1, o2, o3, o4 being object of C for f being Morphism of o1,o2 for g being Morphism of o2,o3 for h being Morphism of o3,o4 st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o4^> <> {} holds (h * g) * f = h * (g * f) );
d9_altcat_2:: for C being non empty AltCatStr holds ( C is with_units iff for o being object of C holds ( <^o,o^> <> {} & ex i being Morphism of o,o st for o9 being object of C for m9 being Morphism of o9,o for m99 being Morphism of o,o9 holds ( ( <^o9,o^> <> {} implies i * m9 = m9 ) & ( <^o,o9^> <> {} implies m99 * i = m99 ) ) ) );
s1_altcat_2:: scheme OnSingletons{ F1() -> non empty set , F2( set ) -> set , P1[ set ] } : { [o,F2(o)] where o is Element of F1() : P1[o] } is Function
s2_altcat_2:: scheme DomOnSingletons{ F1() -> non empty set , F2() -> Function, F3( set ) -> set , P1[ set ] } : dom F2() = { o where o is Element of F1() : P1[o] } provided A1: F2() = { [o,F3(o)] where o is Element of F1() : P1[o] }
s3_altcat_2:: scheme ValOnSingletons{ F1() -> non empty set , F2() -> Function, F3() -> Element of F1(), F4( set ) -> set , P1[ set ] } : F2() . F3() = F4(F3()) provided A1: F2() = { [o,F4(o)] where o is Element of F1() : P1[o] } and A2: P1[F3()]
t1_altcat_2:: for X1, X2, a1, a2 being set holds [:(X1 --> a1),(X2 --> a2):] = [:X1,X2:] --> [a1,a2]
t10_altcat_2:: for C being Category for i, j, k being Object of C holds [:(Hom (j,k)),(Hom (i,j)):] c= dom the Comp of C
t11_altcat_2:: for C being Category for i, j, k being Object of C holds the Comp of C .: [:(Hom (j,k)),(Hom (i,j)):] c= Hom (i,k)
t12_altcat_2:: for C being Category for i being Object of C holds id i in (the_hom_sets_of C) . (i,i)
t13_altcat_2:: for C being Category for i, j, k being Object of C st Hom (i,j) <> {} & Hom (j,k) <> {} holds for f being Morphism of i,j for g being Morphism of j,k holds ((the_comps_of C) . (i,j,k)) . (g,f) = g * f
t14_altcat_2:: for C being Category holds the_comps_of C is associative
t15_altcat_2:: for C being Category holds ( the_comps_of C is with_left_units & the_comps_of C is with_right_units )
t16_altcat_2:: for C being Category holds Alter C is associative
t17_altcat_2:: for C being Category holds Alter C is with_units
t18_altcat_2:: for C being Category holds Alter C is transitive
t19_altcat_2:: for E being empty strict AltCatStr holds E = the_empty_category by Lm1;
t2_altcat_2:: for f, g being Function holds ~ (g * f) = g * (~ f)
t20_altcat_2:: for C being AltCatStr holds C is SubCatStr of C
t21_altcat_2:: for C1, C2, C3 being AltCatStr st C1 is SubCatStr of C2 & C2 is SubCatStr of C3 holds C1 is SubCatStr of C3
t22_altcat_2:: for C1, C2 being AltCatStr st C1 is SubCatStr of C2 & C2 is SubCatStr of C1 holds AltCatStr(# the carrier of C1, the Arrows of C1, the Comp of C1 #) = AltCatStr(# the carrier of C2, the Arrows of C2, the Comp of C2 #)
t23_altcat_2:: for C being non empty AltCatStr for o being object of C for o9 being object of (ObCat o) holds o9 = o
t24_altcat_2:: for C being non empty transitive AltCatStr for D1, D2 being non empty transitive SubCatStr of C st the carrier of D1 c= the carrier of D2 & the Arrows of D1 cc= the Arrows of D2 holds D1 is SubCatStr of D2
t25_altcat_2:: for C being non empty transitive AltCatStr for D being SubCatStr of C st the carrier of D = the carrier of C & the Arrows of D = the Arrows of C holds AltCatStr(# the carrier of D, the Arrows of D, the Comp of D #) = AltCatStr(# the carrier of C, the Arrows of C, the Comp of C #)
t26_altcat_2:: for C being non empty transitive AltCatStr for D1, D2 being non empty transitive SubCatStr of C st the carrier of D1 = the carrier of D2 & the Arrows of D1 = the Arrows of D2 holds AltCatStr(# the carrier of D1, the Arrows of D1, the Comp of D1 #) = AltCatStr(# the carrier of D2, the Arrows of D2, the Comp of D2 #)
t27_altcat_2:: for C being non empty transitive AltCatStr for D being full SubCatStr of C st the carrier of D = the carrier of C holds AltCatStr(# the carrier of D, the Arrows of D, the Comp of D #) = AltCatStr(# the carrier of C, the Arrows of C, the Comp of C #)
t28_altcat_2:: for C being non empty AltCatStr for D being non empty full SubCatStr of C for o1, o2 being object of C for p1, p2 being object of D st o1 = p1 & o2 = p2 holds <^o1,o2^> = <^p1,p2^>
t29_altcat_2:: for C being non empty AltCatStr for D being non empty SubCatStr of C for o being object of D holds o is object of C
t3_altcat_2:: for f, g, h being Function holds ~ (f * [:g,h:]) = (~ f) * [:h,g:]
t30_altcat_2:: for C being non empty transitive AltCatStr for D1, D2 being non empty full SubCatStr of C st the carrier of D1 = the carrier of D2 holds AltCatStr(# the carrier of D1, the Arrows of D1, the Comp of D1 #) = AltCatStr(# the carrier of D2, the Arrows of D2, the Comp of D2 #)
t31_altcat_2:: for C being non empty AltCatStr for D being non empty SubCatStr of C for o1, o2 being object of C for p1, p2 being object of D st o1 = p1 & o2 = p2 holds <^p1,p2^> c= <^o1,o2^>
t32_altcat_2:: for C being non empty transitive AltCatStr for D being non empty transitive SubCatStr of C for p1, p2, p3 being object of D st <^p1,p2^> <> {} & <^p2,p3^> <> {} holds for o1, o2, o3 being object of C st o1 = p1 & o2 = p2 & o3 = p3 holds for f being Morphism of o1,o2 for g being Morphism of o2,o3 for ff being Morphism of p1,p2 for gg being Morphism of p2,p3 st f = ff & g = gg holds g * f = gg * ff
t33_altcat_2:: for C being non empty AltCatStr for D being non empty SubCatStr of C for o1, o2 being object of C for p1, p2 being object of D st o1 = p1 & o2 = p2 & <^p1,p2^> <> {} holds for n being Morphism of p1,p2 holds n is Morphism of o1,o2
t34_altcat_2:: for C being category for D being non empty subcategory of C for o being object of D for o9 being object of C st o = o9 holds idm o = idm o9
t4_altcat_2:: for I being set for A, B, C being ManySortedSet of I st A is_transformable_to B holds for F being ManySortedFunction of A,B for G being ManySortedFunction of B,C holds G ** F is ManySortedFunction of A,C
t5_altcat_2:: for I1 being set for I2 being non empty set for f being Function of I1,I2 for B, C being ManySortedSet of I2 for G being ManySortedFunction of B,C holds G * f is ManySortedFunction of B * f,C * f
t6_altcat_2:: for I1, I2 being non empty set for M being ManySortedSet of [:I1,I2:] for o1 being Element of I1 for o2 being Element of I2 holds (~ M) . (o2,o1) = M . (o1,o2)
t7_altcat_2:: for I, J being set for A being ManySortedSet of I for B being ManySortedSet of J st A cc= B & B cc= A holds A = B
t8_altcat_2:: for I, J, K being set for A being ManySortedSet of I for B being ManySortedSet of J for C being ManySortedSet of K st A cc= B & B cc= C holds A cc= C
t9_altcat_2:: for I being set for A, B being ManySortedSet of I holds ( A cc= B iff A c= B )
d1_altcat_3:: for C being non empty with_units AltCatStr for o1, o2 being object of C for A being Morphism of o1,o2 for B being Morphism of o2,o1 holds ( A is_left_inverse_of B iff A * B = idm o2 );
d10_altcat_3:: for C being AltGraph for o being object of C holds ( o is terminal iff for o1 being object of C ex M being Morphism of o1,o st ( M in <^o1,o^> & <^o1,o^> is trivial ) );
d11_altcat_3:: for C being AltGraph for o being object of C holds ( o is _zero iff ( o is initial & o is terminal ) );
d12_altcat_3:: for C being non empty AltCatStr for o1, o2 being object of C for M being Morphism of o1,o2 holds ( M is _zero iff for o being object of C st o is _zero holds for A being Morphism of o1,o for B being Morphism of o,o2 holds M = B * A );
d2_altcat_3:: for C being non empty with_units AltCatStr for o1, o2 being object of C for A being Morphism of o1,o2 holds ( A is retraction iff ex B being Morphism of o2,o1 st B is_right_inverse_of A );
d3_altcat_3:: for C being non empty with_units AltCatStr for o1, o2 being object of C for A being Morphism of o1,o2 holds ( A is coretraction iff ex B being Morphism of o2,o1 st B is_left_inverse_of A );
d4_altcat_3:: for C being category for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds for A being Morphism of o1,o2 st A is retraction & A is coretraction holds for b5 being Morphism of o2,o1 holds ( b5 = A " iff ( b5 is_left_inverse_of A & b5 is_right_inverse_of A ) );
d5_altcat_3:: for C being category for o1, o2 being object of C for A being Morphism of o1,o2 holds ( A is iso iff ( A * (A ") = idm o2 & (A ") * A = idm o1 ) );
d6_altcat_3:: for C being category for o1, o2 being object of C holds ( o1,o2 are_iso iff ( <^o1,o2^> <> {} & <^o2,o1^> <> {} & ex A being Morphism of o1,o2 st A is iso ) );
d7_altcat_3:: for C being non empty AltCatStr for o1, o2 being object of C for A being Morphism of o1,o2 holds ( A is mono iff for o being object of C st <^o,o1^> <> {} holds for B, C being Morphism of o,o1 st A * B = A * C holds B = C );
d8_altcat_3:: for C being non empty AltCatStr for o1, o2 being object of C for A being Morphism of o1,o2 holds ( A is epi iff for o being object of C st <^o2,o^> <> {} holds for B, C being Morphism of o2,o st B * A = C * A holds B = C );
d9_altcat_3:: for C being AltGraph for o being object of C holds ( o is initial iff for o1 being object of C ex M being Morphism of o,o1 st ( M in <^o,o1^> & <^o,o1^> is trivial ) );
t1_altcat_3:: for C being non empty with_units AltCatStr for o being object of C holds ( idm o is retraction & idm o is coretraction )
t10_altcat_3:: for C being non empty transitive associative AltCatStr for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds for A being Morphism of o1,o2 for B being Morphism of o2,o3 st A is epi & B is epi holds B * A is epi
t11_altcat_3:: for C being non empty transitive associative AltCatStr for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds for A being Morphism of o1,o2 for B being Morphism of o2,o3 st B * A is mono holds A is mono
t12_altcat_3:: for C being non empty transitive associative AltCatStr for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds for A being Morphism of o1,o2 for B being Morphism of o2,o3 st B * A is epi holds B is epi
t13_altcat_3:: for X being non empty set for o1, o2 being object of (EnsCat X) st <^o1,o2^> <> {} holds for A being Morphism of o1,o2 for F being Function of o1,o2 st F = A holds ( A is mono iff F is one-to-one )
t14_altcat_3:: for X being non empty with_non-empty_elements set for o1, o2 being object of (EnsCat X) st <^o1,o2^> <> {} holds for A being Morphism of o1,o2 for F being Function of o1,o2 st F = A holds ( A is epi iff F is onto )
t15_altcat_3:: for C being category for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds for A being Morphism of o1,o2 st A is retraction holds A is epi
t16_altcat_3:: for C being category for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds for A being Morphism of o1,o2 st A is coretraction holds A is mono
t17_altcat_3:: for C being category for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds for A being Morphism of o1,o2 st A is iso holds ( A is mono & A is epi )
t18_altcat_3:: for C being category for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} holds for A being Morphism of o1,o2 for B being Morphism of o2,o3 st A is retraction & B is retraction holds B * A is retraction
t19_altcat_3:: for C being category for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} holds for A being Morphism of o1,o2 for B being Morphism of o2,o3 st A is coretraction & B is coretraction holds B * A is coretraction
t2_altcat_3:: for C being category for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds for A being Morphism of o1,o2 st A is retraction & A is coretraction holds ( (A ") * A = idm o1 & A * (A ") = idm o2 )
t20_altcat_3:: for C being category for o1, o2 being object of C for A being Morphism of o1,o2 st A is retraction & A is mono & <^o1,o2^> <> {} & <^o2,o1^> <> {} holds A is iso
t21_altcat_3:: for C being category for o1, o2 being object of C for A being Morphism of o1,o2 st A is coretraction & A is epi & <^o1,o2^> <> {} & <^o2,o1^> <> {} holds A is iso
t22_altcat_3:: for C being category for o1, o2, o3 being object of C for A being Morphism of o1,o2 for B being Morphism of o2,o3 st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} & B * A is retraction holds B is retraction
t23_altcat_3:: for C being category for o1, o2, o3 being object of C for A being Morphism of o1,o2 for B being Morphism of o2,o3 st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} & B * A is coretraction holds A is coretraction
t24_altcat_3:: for C being category st ( for o1, o2 being object of C for A1 being Morphism of o1,o2 holds A1 is retraction ) holds for a, b being object of C for A being Morphism of a,b st <^a,b^> <> {} & <^b,a^> <> {} holds A is iso
t25_altcat_3:: for C being AltGraph for o being object of C holds ( o is initial iff for o1 being object of C ex M being Morphism of o,o1 st ( M in <^o,o1^> & ( for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds M = M1 ) ) )
t26_altcat_3:: for C being category for o1, o2 being object of C st o1 is initial & o2 is initial holds o1,o2 are_iso
t27_altcat_3:: for C being AltGraph for o being object of C holds ( o is terminal iff for o1 being object of C ex M being Morphism of o1,o st ( M in <^o1,o^> & ( for M1 being Morphism of o1,o st M1 in <^o1,o^> holds M = M1 ) ) )
t28_altcat_3:: for C being category for o1, o2 being object of C st o1 is terminal & o2 is terminal holds o1,o2 are_iso
t29_altcat_3:: for C being category for o1, o2 being object of C st o1 is _zero & o2 is _zero holds o1,o2 are_iso
t3_altcat_3:: for C being category for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds for A being Morphism of o1,o2 st A is retraction & A is coretraction holds (A ") " = A
t30_altcat_3:: for C being category for o1, o2, o3 being object of C for M1 being Morphism of o1,o2 for M2 being Morphism of o2,o3 st M1 is _zero & M2 is _zero holds M2 * M1 is _zero
t4_altcat_3:: for C being category for o being object of C holds (idm o) " = idm o
t5_altcat_3:: for C being category for o1, o2 being object of C for A being Morphism of o1,o2 st A is iso holds ( A is retraction & A is coretraction )
t6_altcat_3:: for C being category for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds for A being Morphism of o1,o2 holds ( A is iso iff ( A is retraction & A is coretraction ) )
t7_altcat_3:: for C being category for o1, o2, o3 being object of C for A being Morphism of o1,o2 for B being Morphism of o2,o3 st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} & A is iso & B is iso holds ( B * A is iso & (B * A) " = (A ") * (B ") )
t8_altcat_3:: for C being category for o1, o2, o3 being object of C st o1,o2 are_iso & o2,o3 are_iso holds o1,o3 are_iso
t9_altcat_3:: for C being non empty transitive associative AltCatStr for o1, o2, o3 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds for A being Morphism of o1,o2 for B being Morphism of o2,o3 st A is mono & B is mono holds B * A is mono
d1_altcat_4:: for C being category for b2 being non empty transitive strict SubCatStr of C holds ( b2 = AllMono C iff ( the carrier of b2 = the carrier of C & the Arrows of b2 cc= the Arrows of C & ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m in the Arrows of b2 . (o1,o2) iff ( <^o1,o2^> <> {} & m is mono ) ) ) ) );
d2_altcat_4:: for C being category for b2 being non empty transitive strict SubCatStr of C holds ( b2 = AllEpi C iff ( the carrier of b2 = the carrier of C & the Arrows of b2 cc= the Arrows of C & ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m in the Arrows of b2 . (o1,o2) iff ( <^o1,o2^> <> {} & m is epi ) ) ) ) );
d3_altcat_4:: for C being category for b2 being non empty transitive strict SubCatStr of C holds ( b2 = AllRetr C iff ( the carrier of b2 = the carrier of C & the Arrows of b2 cc= the Arrows of C & ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m in the Arrows of b2 . (o1,o2) iff ( <^o1,o2^> <> {} & <^o2,o1^> <> {} & m is retraction ) ) ) ) );
d4_altcat_4:: for C being category for b2 being non empty transitive strict SubCatStr of C holds ( b2 = AllCoretr C iff ( the carrier of b2 = the carrier of C & the Arrows of b2 cc= the Arrows of C & ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m in the Arrows of b2 . (o1,o2) iff ( <^o1,o2^> <> {} & <^o2,o1^> <> {} & m is coretraction ) ) ) ) );
d5_altcat_4:: for C being category for b2 being non empty transitive strict SubCatStr of C holds ( b2 = AllIso C iff ( the carrier of b2 = the carrier of C & the Arrows of b2 cc= the Arrows of C & ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m in the Arrows of b2 . (o1,o2) iff ( <^o1,o2^> <> {} & <^o2,o1^> <> {} & m is iso ) ) ) ) );
t1_altcat_4:: for C being category for o1, o2, o3 being object of C for v being Morphism of o1,o2 for u being Morphism of o1,o3 for f being Morphism of o2,o3 st u = f * v & (f ") * f = idm o2 & <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o2^> <> {} holds v = (f ") * u
t10_altcat_4:: for C being category for o2, o1 being object of C st o2 is terminal & o1,o2 are_iso holds o1 is terminal
t11_altcat_4:: for C being category for o1, o2 being object of C st o1 is initial & o1,o2 are_iso holds o2 is initial
t12_altcat_4:: for C being category for o1, o2 being object of C st o1 is initial & o2 is terminal & <^o2,o1^> <> {} holds ( o2 is initial & o1 is terminal )
t13_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being contravariant Functor of A,B for a being object of A holds F . (idm a) = idm (F . a)
t14_altcat_4:: for C1, C2 being non empty AltCatStr for F being Contravariant FunctorStr over C1,C2 holds ( F is full iff for o1, o2 being object of C1 holds Morph-Map (F,o2,o1) is onto )
t15_altcat_4:: for C1, C2 being non empty AltCatStr for F being Contravariant FunctorStr over C1,C2 holds ( F is faithful iff for o1, o2 being object of C1 holds Morph-Map (F,o2,o1) is one-to-one )
t16_altcat_4:: for C1, C2 being non empty AltCatStr for F being Covariant FunctorStr over C1,C2 for o1, o2 being object of C1 for Fm being Morphism of (F . o1),(F . o2) st <^o1,o2^> <> {} & F is full & F is feasible holds ex m being Morphism of o1,o2 st Fm = F . m
t17_altcat_4:: for C1, C2 being non empty AltCatStr for F being Contravariant FunctorStr over C1,C2 for o1, o2 being object of C1 for Fm being Morphism of (F . o2),(F . o1) st <^o1,o2^> <> {} & F is full & F is feasible holds ex m being Morphism of o1,o2 st Fm = F . m
t18_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being covariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds F . a is retraction
t19_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being covariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is coretraction holds F . a is coretraction
t2_altcat_4:: for C being category for o2, o3, o1 being object of C for v being Morphism of o2,o3 for u being Morphism of o1,o3 for f being Morphism of o1,o2 st u = v * f & f * (f ") = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} holds v = u * (f ")
t20_altcat_4:: for A, B being category for F being covariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is iso holds F . a is iso
t21_altcat_4:: for A, B being category for F being covariant Functor of A,B for o1, o2 being object of A st o1,o2 are_iso holds F . o1,F . o2 are_iso
t22_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being contravariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is retraction holds F . a is coretraction
t23_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being contravariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is coretraction holds F . a is retraction
t24_altcat_4:: for A, B being category for F being contravariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is iso holds F . a is iso
t25_altcat_4:: for A, B being category for F being contravariant Functor of A,B for o1, o2 being object of A st o1,o2 are_iso holds F . o2,F . o1 are_iso
t26_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being covariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is retraction holds a is retraction
t27_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being covariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is coretraction holds a is coretraction
t28_altcat_4:: for A, B being category for F being covariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is iso holds a is iso
t29_altcat_4:: for A, B being category for F being covariant Functor of A,B for o1, o2 being object of A st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . o1,F . o2 are_iso holds o1,o2 are_iso
t3_altcat_4:: for C being category for o1, o2 being object of C for m being Morphism of o1,o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & m is iso holds m " is iso
t30_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being contravariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is retraction holds a is coretraction
t31_altcat_4:: for A, B being non empty transitive with_units AltCatStr for F being contravariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is coretraction holds a is retraction
t32_altcat_4:: for A, B being category for F being contravariant Functor of A,B for o1, o2 being object of A for a being Morphism of o1,o2 st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . a is iso holds a is iso
t33_altcat_4:: for A, B being category for F being contravariant Functor of A,B for o1, o2 being object of A st F is full & F is faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {} & F . o2,F . o1 are_iso holds o1,o2 are_iso
t34_altcat_4:: for C being AltCatStr for D being SubCatStr of C st the carrier of C = the carrier of D & the Arrows of C = the Arrows of D holds D is full
t35_altcat_4:: for C being non empty with_units AltCatStr for D being SubCatStr of C st the carrier of C = the carrier of D & the Arrows of C = the Arrows of D holds D is id-inheriting
t36_altcat_4:: for C being category for B being non empty subcategory of C for A being non empty subcategory of B holds A is non empty subcategory of C
t37_altcat_4:: for C being non empty transitive AltCatStr for D being non empty transitive SubCatStr of C for o1, o2 being object of C for p1, p2 being object of D for m being Morphism of o1,o2 for n being Morphism of p1,p2 st p1 = o1 & p2 = o2 & m = n & <^p1,p2^> <> {} holds ( ( m is mono implies n is mono ) & ( m is epi implies n is epi ) )
t38_altcat_4:: for C being category for D being non empty subcategory of C for o1, o2 being object of C for p1, p2 being object of D for m being Morphism of o1,o2 for m1 being Morphism of o2,o1 for n being Morphism of p1,p2 for n1 being Morphism of p2,p1 st p1 = o1 & p2 = o2 & m = n & m1 = n1 & <^p1,p2^> <> {} & <^p2,p1^> <> {} holds ( ( m is_left_inverse_of m1 implies n is_left_inverse_of n1 ) & ( n is_left_inverse_of n1 implies m is_left_inverse_of m1 ) & ( m is_right_inverse_of m1 implies n is_right_inverse_of n1 ) & ( n is_right_inverse_of n1 implies m is_right_inverse_of m1 ) )
t39_altcat_4:: for C being category for D being non empty full subcategory of C for o1, o2 being object of C for p1, p2 being object of D for m being Morphism of o1,o2 for n being Morphism of p1,p2 st p1 = o1 & p2 = o2 & m = n & <^p1,p2^> <> {} & <^p2,p1^> <> {} holds ( ( m is retraction implies n is retraction ) & ( m is coretraction implies n is coretraction ) & ( m is iso implies n is iso ) )
t4_altcat_4:: for C being non empty with_units AltCatStr for o being object of C holds ( idm o is epi & idm o is mono )
t40_altcat_4:: for C being category for D being non empty subcategory of C for o1, o2 being object of C for p1, p2 being object of D for m being Morphism of o1,o2 for n being Morphism of p1,p2 st p1 = o1 & p2 = o2 & m = n & <^p1,p2^> <> {} & <^p2,p1^> <> {} holds ( ( n is retraction implies m is retraction ) & ( n is coretraction implies m is coretraction ) & ( n is iso implies m is iso ) )
t41_altcat_4:: for C being category holds AllIso C is non empty subcategory of AllRetr C
t42_altcat_4:: for C being category holds AllIso C is non empty subcategory of AllCoretr C
t43_altcat_4:: for C being category holds AllCoretr C is non empty subcategory of AllMono C
t44_altcat_4:: for C being category holds AllRetr C is non empty subcategory of AllEpi C
t45_altcat_4:: for C being category st ( for o1, o2 being object of C for m being Morphism of o1,o2 holds m is mono ) holds AltCatStr(# the carrier of C, the Arrows of C, the Comp of C #) = AllMono C
t46_altcat_4:: for C being category st ( for o1, o2 being object of C for m being Morphism of o1,o2 holds m is epi ) holds AltCatStr(# the carrier of C, the Arrows of C, the Comp of C #) = AllEpi C
t47_altcat_4:: for C being category st ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m is retraction & <^o2,o1^> <> {} ) ) holds AltCatStr(# the carrier of C, the Arrows of C, the Comp of C #) = AllRetr C
t48_altcat_4:: for C being category st ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m is coretraction & <^o2,o1^> <> {} ) ) holds AltCatStr(# the carrier of C, the Arrows of C, the Comp of C #) = AllCoretr C
t49_altcat_4:: for C being category st ( for o1, o2 being object of C for m being Morphism of o1,o2 holds ( m is iso & <^o2,o1^> <> {} ) ) holds AltCatStr(# the carrier of C, the Arrows of C, the Comp of C #) = AllIso C
t5_altcat_4:: for C being category for o1, o2 being object of C for f being Morphism of o1,o2 for g, h being Morphism of o2,o1 st h * f = idm o1 & f * g = idm o2 & <^o1,o2^> <> {} & <^o2,o1^> <> {} holds g = h
t50_altcat_4:: for C being category for o1, o2 being object of (AllMono C) for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds m is mono
t51_altcat_4:: for C being category for o1, o2 being object of (AllEpi C) for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds m is epi
t52_altcat_4:: for C being category for o1, o2 being object of (AllIso C) for m being Morphism of o1,o2 st <^o1,o2^> <> {} holds ( m is iso & m " in <^o2,o1^> )
t53_altcat_4:: for C being category holds AllMono (AllMono C) = AllMono C
t54_altcat_4:: for C being category holds AllEpi (AllEpi C) = AllEpi C
t55_altcat_4:: for C being category holds AllIso (AllIso C) = AllIso C
t56_altcat_4:: for C being category holds AllIso (AllMono C) = AllIso C
t57_altcat_4:: for C being category holds AllIso (AllEpi C) = AllIso C
t58_altcat_4:: for C being category holds AllIso (AllRetr C) = AllIso C
t59_altcat_4:: for C being category holds AllIso (AllCoretr C) = AllIso C
t6_altcat_4:: for C being category st ( for o1, o2 being object of C for f being Morphism of o1,o2 holds f is coretraction ) holds for a, b being object of C for g being Morphism of a,b st <^a,b^> <> {} & <^b,a^> <> {} holds g is iso
t7_altcat_4:: for C being category for o1, o2 being object of C for m, m9 being Morphism of o1,o2 st m is _zero & m9 is _zero & ex O being object of C st O is _zero holds m = m9
t8_altcat_4:: for C being non empty AltCatStr for O, A being object of C for M being Morphism of O,A st O is terminal holds M is mono
t9_altcat_4:: for C being non empty AltCatStr for O, A being object of C for M being Morphism of A,O st O is initial holds M is epi
d1_ami_2:: SCM-Memory = {NAT} \/ SCM-Data-Loc;
d10_ami_2:: canceled;
d11_ami_2:: canceled;
d12_ami_2:: canceled;
d13_ami_2:: canceled;
d14_ami_2:: for x being Element of SCM-Instr for s being SCM-State holds ( ( ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),(succ (IC s))) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),(succ (IC s))) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),(succ (IC s))) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),(succ (IC s))) ) & ( ex mk, ml being Element of SCM-Data-Loc st x = [5,{},<*mk,ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg ((SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) div (s . (x address_2))))),(x address_2),((s . (x address_1)) mod (s . (x address_2))))),(succ (IC s))) ) & ( ex mk being Element of NAT st x = [6,<*mk*>,{}] implies SCM-Exec-Res (x,s) = SCM-Chg (s,(x jump_address)) ) & ( ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg (s,(IFEQ ((s . (x cond_address)),0,(x cjump_address),(succ (IC s))))) ) & ( ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [8,<*mk*>,<*ml*>] implies SCM-Exec-Res (x,s) = SCM-Chg (s,(IFGT ((s . (x cond_address)),0,(x cjump_address),(succ (IC s))))) ) & ( ( for mk, ml being Element of SCM-Data-Loc holds not x = [1,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [2,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [3,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [4,{},<*mk,ml*>] ) & ( for mk, ml being Element of SCM-Data-Loc holds not x = [5,{},<*mk,ml*>] ) & ( for mk being Element of NAT holds not x = [6,<*mk*>,{}] ) & ( for mk being Element of NAT for ml being Element of SCM-Data-Loc holds not x = [7,<*mk*>,<*ml*>] ) & ( for mk being Element of NAT for ml being Element of SCM-Data-Loc holds not x = [8,<*mk*>,<*ml*>] ) implies SCM-Exec-Res (x,s) = s ) );
d15_ami_2:: for b1 being Action of SCM-Instr,(product (SCM-VAL * SCM-OK)) holds ( b1 = SCM-Exec iff for x being Element of SCM-Instr for y being SCM-State holds (b1 . x) . y = SCM-Exec-Res (x,y) );
d16_ami_2:: for x being set holds ( x is Int-like iff x in SCM-Data-Loc );
d2_ami_2:: canceled;
d3_ami_2:: canceled;
d4_ami_2:: for b1 being Function of SCM-Memory,2 holds ( b1 = SCM-OK iff for k being Element of SCM-Memory holds ( ( k = NAT implies b1 . k = 0 ) & ( k in SCM-Data-Loc implies b1 . k = 1 ) ) );
d5_ami_2:: SCM-VAL = <%NAT,INT%>;
d6_ami_2:: for s being SCM-State holds IC s = s . NAT;
d7_ami_2:: for s being SCM-State for u being Nat holds SCM-Chg (s,u) = s +* (NAT .--> u);
d8_ami_2:: for s being SCM-State for t being Element of SCM-Data-Loc for u being Integer holds SCM-Chg (s,t,u) = s +* (t .--> u);
d9_ami_2:: canceled;
t1_ami_2:: canceled;
t10_ami_2:: for a being Element of SCM-Data-Loc holds pi ((product (SCM-VAL * SCM-OK)),a) = INT
t11_ami_2:: for s being SCM-State for u being Nat holds (SCM-Chg (s,u)) . NAT = u
t12_ami_2:: for s being SCM-State for u being Nat for mk being Element of SCM-Data-Loc holds (SCM-Chg (s,u)) . mk = s . mk
t13_ami_2:: for s being SCM-State for u, v being Nat holds (SCM-Chg (s,u)) . v = s . v
t14_ami_2:: for s being SCM-State for t being Element of SCM-Data-Loc for u being Integer holds (SCM-Chg (s,t,u)) . NAT = s . NAT
t15_ami_2:: for s being SCM-State for t being Element of SCM-Data-Loc for u being Integer holds (SCM-Chg (s,t,u)) . t = u
t16_ami_2:: for s being SCM-State for t being Element of SCM-Data-Loc for u being Integer for mk being Element of SCM-Data-Loc st mk <> t holds (SCM-Chg (s,t,u)) . mk = s . mk
t17_ami_2:: canceled;
t18_ami_2:: canceled;
t19_ami_2:: canceled;
t2_ami_2:: canceled;
t20_ami_2:: not NAT in SCM-Data-Loc by Lm2;
t21_ami_2:: canceled;
t22_ami_2:: NAT in SCM-Memory by Lm3;
t23_ami_2:: for x being set st x in SCM-Data-Loc holds ex k being Element of NAT st x = [1,k]
t24_ami_2:: for k being Nat holds [1,k] in SCM-Data-Loc
t25_ami_2:: canceled;
t26_ami_2:: for k being Element of SCM-Memory holds ( k = NAT or k in SCM-Data-Loc ) by Lm1;
t27_ami_2:: dom (SCM-VAL * SCM-OK) = SCM-Memory by Lm5;
t28_ami_2:: for s being SCM-State holds dom s = SCM-Memory by Lm5, CARD_3:9;
t3_ami_2:: canceled;
t4_ami_2:: canceled;
t5_ami_2:: canceled;
t6_ami_2:: (SCM-VAL * SCM-OK) . NAT = NAT
t7_ami_2:: for i being Element of SCM-Memory st i in SCM-Data-Loc holds (SCM-VAL * SCM-OK) . i = INT
t8_ami_2:: for a being Element of SCM-Data-Loc holds (SCM-VAL * SCM-OK) . a = INT by Th7;
t9_ami_2:: pi ((product (SCM-VAL * SCM-OK)),NAT) = NAT by Th6, Lm5, Lm3, CARD_3:12;
d1_ami_3:: SCM = AMI-Struct(# SCM-Memory,(In (NAT,SCM-Memory)),SCM-Instr,SCM-OK,SCM-VAL,SCM-Exec #);
d10_ami_3:: for loc being Nat for a being Data-Location holds a >0_goto loc = [8,<*loc*>,<*a*>];
d11_ami_3:: for k being Nat holds dl. k = [1,k];
d2_ami_3:: canceled;
d3_ami_3:: for a, b being Data-Location holds a := b = [1,{},<*a,b*>];
d4_ami_3:: for a, b being Data-Location holds AddTo (a,b) = [2,{},<*a,b*>];
d5_ami_3:: for a, b being Data-Location holds SubFrom (a,b) = [3,{},<*a,b*>];
d6_ami_3:: for a, b being Data-Location holds MultBy (a,b) = [4,{},<*a,b*>];
d7_ami_3:: for a, b being Data-Location holds Divide (a,b) = [5,{},<*a,b*>];
d8_ami_3:: for loc being Nat holds SCM-goto loc = [6,<*loc*>,{}];
d9_ami_3:: for loc being Nat for a being Data-Location holds a =0_goto loc = [7,<*loc*>,<*a*>];
t1_ami_3:: IC = NAT by AMI_2:22, FUNCT_7:def_1;
t10_ami_3:: for i, j being Nat st i <> j holds dl. i <> dl. j by XTUPLE_0:1;
t11_ami_3:: for l being Data-Location holds Values l = INT
t12_ami_3:: for i, j being Nat holds dl. i <> j ;
t13_ami_3:: for i being Nat holds ( IC <> dl. i & IC <> i )
t14_ami_3:: for I being Instruction of SCM st ex s being State of SCM st (Exec (I,s)) . (IC ) = succ (IC s) holds not I is halting by Lm2;
t15_ami_3:: for I being Instruction of SCM st I = [0,{},{}] holds I is halting by Lm3;
t16_ami_3:: for a, b being Data-Location holds not a := b is halting by Lm4;
t17_ami_3:: for a, b being Data-Location holds not AddTo (a,b) is halting by Lm5;
t18_ami_3:: for a, b being Data-Location holds not SubFrom (a,b) is halting by Lm6;
t19_ami_3:: for a, b being Data-Location holds not MultBy (a,b) is halting by Lm7;
t2_ami_3:: for a, b being Data-Location for s being State of SCM holds ( (Exec ((a := b),s)) . (IC ) = succ (IC s) & (Exec ((a := b),s)) . a = s . b & ( for c being Data-Location st c <> a holds (Exec ((a := b),s)) . c = s . c ) )
t20_ami_3:: for a, b being Data-Location holds not Divide (a,b) is halting by Lm8;
t21_ami_3:: for loc being Nat holds not SCM-goto loc is halting by Lm9;
t22_ami_3:: for a being Data-Location for loc being Nat holds not a =0_goto loc is halting by Lm10;
t23_ami_3:: for a being Data-Location for loc being Nat holds not a >0_goto loc is halting by Lm11;
t24_ami_3:: for I being set holds ( I is Instruction of SCM iff ( I = [0,{},{}] or ex a, b being Data-Location st I = a := b or ex a, b being Data-Location st I = AddTo (a,b) or ex a, b being Data-Location st I = SubFrom (a,b) or ex a, b being Data-Location st I = MultBy (a,b) or ex a, b being Data-Location st I = Divide (a,b) or ex loc being Nat st I = SCM-goto loc or ex a being Data-Location ex loc being Nat st I = a =0_goto loc or ex a being Data-Location ex loc being Nat st I = a >0_goto loc ) ) by Lm12;
t25_ami_3:: for I being Instruction of SCM st I is halting holds I = halt SCM by Lm13;
t26_ami_3:: halt SCM = [0,{},{}] ;
t27_ami_3:: Data-Locations = SCM-Data-Loc
t28_ami_3:: for d being Data-Location holds d in Data-Locations by Th27, AMI_2:def_16;
t29_ami_3:: for s being SCM-State holds s is State of SCM by Lm1;
t3_ami_3:: for a, b being Data-Location for s being State of SCM holds ( (Exec ((AddTo (a,b)),s)) . (IC ) = succ (IC s) & (Exec ((AddTo (a,b)),s)) . a = (s . a) + (s . b) & ( for c being Data-Location st c <> a holds (Exec ((AddTo (a,b)),s)) . c = s . c ) )
t30_ami_3:: for l being Element of SCM-Instr holds InsCode l <= 8 by SCM_INST:10;
t4_ami_3:: for a, b being Data-Location for s being State of SCM holds ( (Exec ((SubFrom (a,b)),s)) . (IC ) = succ (IC s) & (Exec ((SubFrom (a,b)),s)) . a = (s . a) - (s . b) & ( for c being Data-Location st c <> a holds (Exec ((SubFrom (a,b)),s)) . c = s . c ) )
t5_ami_3:: for a, b being Data-Location for s being State of SCM holds ( (Exec ((MultBy (a,b)),s)) . (IC ) = succ (IC s) & (Exec ((MultBy (a,b)),s)) . a = (s . a) * (s . b) & ( for c being Data-Location st c <> a holds (Exec ((MultBy (a,b)),s)) . c = s . c ) )
t6_ami_3:: for a, b being Data-Location for s being State of SCM holds ( (Exec ((Divide (a,b)),s)) . (IC ) = succ (IC s) & ( a <> b implies (Exec ((Divide (a,b)),s)) . a = (s . a) div (s . b) ) & (Exec ((Divide (a,b)),s)) . b = (s . a) mod (s . b) & ( for c being Data-Location st c <> a & c <> b holds (Exec ((Divide (a,b)),s)) . c = s . c ) )
t7_ami_3:: for c being Data-Location for loc being Nat for s being State of SCM holds ( (Exec ((SCM-goto loc),s)) . (IC ) = loc & (Exec ((SCM-goto loc),s)) . c = s . c )
t8_ami_3:: for a, c being Data-Location for loc being Nat for s being State of SCM holds ( ( s . a = 0 implies (Exec ((a =0_goto loc),s)) . (IC ) = loc ) & ( s . a <> 0 implies (Exec ((a =0_goto loc),s)) . (IC ) = succ (IC s) ) & (Exec ((a =0_goto loc),s)) . c = s . c )
t9_ami_3:: for a, c being Data-Location for loc being Nat for s being State of SCM holds ( ( s . a > 0 implies (Exec ((a >0_goto loc),s)) . (IC ) = loc ) & ( s . a <= 0 implies (Exec ((a >0_goto loc),s)) . (IC ) = succ (IC s) ) & (Exec ((a >0_goto loc),s)) . c = s . c )
d1_ami_4:: Euclide-Algorithm = (0 .--> ((dl. 2) := (dl. 1))) +* ((1 .--> (Divide ((dl. 0),(dl. 1)))) +* ((2 .--> ((dl. 0) := (dl. 2))) +* ((3 .--> ((dl. 1) >0_goto 0)) +* (4 .--> (halt SCM)))));
d2_ami_4:: for b1 being PartFunc of (FinPartSt SCM),(FinPartSt SCM) holds ( b1 = Euclide-Function iff for p, q being FinPartState of SCM holds ( [p,q] in b1 iff ex x, y being Integer st ( x > 0 & y > 0 & p = ((dl. 0),(dl. 1)) --> (x,y) & q = (dl. 0) .--> (x gcd y) ) ) );
t1_ami_4:: dom Euclide-Algorithm = 5
t10_ami_4:: Euclide-Algorithm , Start-At (0,SCM) computes Euclide-Function
t2_ami_4:: for s being State of SCM for P being Instruction-Sequence of SCM st Euclide-Algorithm c= P holds for k being Element of NAT st IC (Comput (P,s,k)) = 0 holds ( IC (Comput (P,s,(k + 1))) = 1 & (Comput (P,s,(k + 1))) . (dl. 0) = (Comput (P,s,k)) . (dl. 0) & (Comput (P,s,(k + 1))) . (dl. 1) = (Comput (P,s,k)) . (dl. 1) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 1) )
t3_ami_4:: for s being State of SCM for P being Instruction-Sequence of SCM st Euclide-Algorithm c= P holds for k being Element of NAT st IC (Comput (P,s,k)) = 1 holds ( IC (Comput (P,s,(k + 1))) = 2 & (Comput (P,s,(k + 1))) . (dl. 0) = ((Comput (P,s,k)) . (dl. 0)) div ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 1) = ((Comput (P,s,k)) . (dl. 0)) mod ((Comput (P,s,k)) . (dl. 1)) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) )
t4_ami_4:: for s being State of SCM for P being Instruction-Sequence of SCM st Euclide-Algorithm c= P holds for k being Element of NAT st IC (Comput (P,s,k)) = 2 holds ( IC (Comput (P,s,(k + 1))) = 3 & (Comput (P,s,(k + 1))) . (dl. 0) = (Comput (P,s,k)) . (dl. 2) & (Comput (P,s,(k + 1))) . (dl. 1) = (Comput (P,s,k)) . (dl. 1) & (Comput (P,s,(k + 1))) . (dl. 2) = (Comput (P,s,k)) . (dl. 2) )
t5_ami_4:: for s being State of SCM for P being Instruction-Sequence of SCM st Euclide-Algorithm c= P holds for k being Element of NAT st IC (Comput (P,s,k)) = 3 holds ( ( (Comput (P,s,k)) . (dl. 1) > 0 implies IC (Comput (P,s,(k + 1))) = 0 ) & ( (Comput (P,s,k)) . (dl. 1) <= 0 implies IC (Comput (P,s,(k + 1))) = 4 ) & (Comput (P,s,(k + 1))) . (dl. 0) = (Comput (P,s,k)) . (dl. 0) & (Comput (P,s,(k + 1))) . (dl. 1) = (Comput (P,s,k)) . (dl. 1) )
t6_ami_4:: for s being State of SCM for P being Instruction-Sequence of SCM st Euclide-Algorithm c= P holds for k, i being Element of NAT st IC (Comput (P,s,k)) = 4 holds Comput (P,s,(k + i)) = Comput (P,s,k)
t7_ami_4:: for s being 0 -started State of SCM for P being Instruction-Sequence of SCM st Euclide-Algorithm c= P holds for x, y being Integer st s . (dl. 0) = x & s . (dl. 1) = y & x > 0 & y > 0 holds (Result (P,s)) . (dl. 0) = x gcd y
t8_ami_4:: for p being set holds ( p in dom Euclide-Function iff ex x, y being Integer st ( x > 0 & y > 0 & p = ((dl. 0),(dl. 1)) --> (x,y) ) )
t9_ami_4:: for i, j being Integer st i > 0 & j > 0 holds Euclide-Function . (((dl. 0),(dl. 1)) --> (i,j)) = (dl. 0) .--> (i gcd j)
t1_ami_5:: for dl being Data-Location ex i being Element of NAT st dl = dl. i
t10_ami_5:: for ins being Instruction of SCM st InsCode ins = 3 holds ex da, db being Data-Location st ins = SubFrom (da,db)
t11_ami_5:: for ins being Instruction of SCM st InsCode ins = 4 holds ex da, db being Data-Location st ins = MultBy (da,db)
t12_ami_5:: for ins being Instruction of SCM st InsCode ins = 5 holds ex da, db being Data-Location st ins = Divide (da,db)
t13_ami_5:: for ins being Instruction of SCM st InsCode ins = 6 holds ex loc being Element of NAT st ins = SCM-goto loc
t14_ami_5:: for ins being Instruction of SCM st InsCode ins = 7 holds ex loc being Element of NAT ex da being Data-Location st ins = da =0_goto loc
t15_ami_5:: for ins being Instruction of SCM st InsCode ins = 8 holds ex loc being Element of NAT ex da being Data-Location st ins = da >0_goto loc
t16_ami_5:: for s being State of SCM for iloc being Element of NAT for a being Data-Location holds s . a = (s +* (Start-At (iloc,SCM))) . a
t17_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da, db being Data-Location for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da := db & da in dom p holds (Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db
t18_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da, db being Data-Location for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds ((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db)
t19_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da, db being Data-Location for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = SubFrom (da,db) & da in dom p holds ((Comput (P1,s1,i)) . da) - ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) - ((Comput (P2,s2,i)) . db)
t2_ami_5:: for dl being Data-Location holds dl <> IC
t20_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da, db being Data-Location for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = MultBy (da,db) & da in dom p holds ((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db)
t21_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da, db being Data-Location for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = Divide (da,db) & da in dom p & da <> db holds ((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db)
t22_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da, db being Data-Location for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = Divide (da,db) & db in dom p holds ((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db)
t23_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da being Data-Location for loc being Element of NAT for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da =0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds ( (Comput (P1,s1,i)) . da = 0 iff (Comput (P2,s2,i)) . da = 0 )
t24_ami_5:: for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function for p being non empty b1 -autonomic FinPartState of SCM for s1, s2 being State of SCM st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds for i being Element of NAT for da being Data-Location for loc being Element of NAT for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> succ (IC (Comput (P1,s1,i))) holds ( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
t25_ami_5:: for s1, s2 being State of SCM st IC s1 = IC s2 & ( for a being Data-Location holds s1 . a = s2 . a ) holds s1 = s2
t3_ami_5:: for il being Element of NAT for dl being Data-Location holds il <> dl
t4_ami_5:: for s being State of SCM for d being Data-Location holds d in dom s
t5_ami_5:: for l being Instruction of SCM holds InsCode l <= 8
t6_ami_5:: canceled;
t7_ami_5:: for ins being Instruction of SCM st InsCode ins = 0 holds ins = halt SCM
t8_ami_5:: for ins being Instruction of SCM st InsCode ins = 1 holds ex da, db being Data-Location st ins = da := db
t9_ami_5:: for ins being Instruction of SCM st InsCode ins = 2 holds ex da, db being Data-Location st ins = AddTo (da,db)
t1_ami_6:: for T being InsType of the InstructionsF of SCM holds ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 or T = 8 )
t10_ami_6:: for T being InsType of the InstructionsF of SCM st T = 7 holds dom (product" (JumpParts T)) = {1}
t11_ami_6:: for T being InsType of the InstructionsF of SCM st T = 8 holds dom (product" (JumpParts T)) = {1}
t12_ami_6:: for k1 being Nat holds (product" (JumpParts (InsCode (SCM-goto k1)))) . 1 = NAT
t13_ami_6:: for a being Data-Location for k1 being Nat holds (product" (JumpParts (InsCode (a =0_goto k1)))) . 1 = NAT
t14_ami_6:: for a being Data-Location for k1 being Nat holds (product" (JumpParts (InsCode (a >0_goto k1)))) . 1 = NAT
t15_ami_6:: for il being Element of NAT for k being Nat holds NIC ((SCM-goto k),il) = {k}
t16_ami_6:: for k being Nat holds JUMP (SCM-goto k) = {k}
t17_ami_6:: for a being Data-Location for il being Element of NAT for k being Nat holds NIC ((a =0_goto k),il) = {k,(succ il)}
t18_ami_6:: for a being Data-Location for k being Nat holds JUMP (a =0_goto k) = {k}
t19_ami_6:: for a being Data-Location for il being Element of NAT for k being Nat holds NIC ((a >0_goto k),il) = {k,(succ il)}
t2_ami_6:: JumpPart (halt SCM) = {} ;
t20_ami_6:: for a being Data-Location for k being Nat holds JUMP (a >0_goto k) = {k}
t21_ami_6:: for il being Element of NAT holds SUCC (il,SCM) = {il,(succ il)}
t22_ami_6:: for k being Element of NAT holds ( k + 1 in SUCC (k,SCM) & ( for j being Element of NAT st j in SUCC (k,SCM) holds k <= j ) )
t23_ami_6:: for i1 being Element of NAT for k being Nat holds IncAddr ((SCM-goto i1),k) = SCM-goto (i1 + k)
t24_ami_6:: for a being Data-Location for i1 being Element of NAT for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
t25_ami_6:: for a being Data-Location for i1 being Element of NAT for k being Nat holds IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k)
t3_ami_6:: for T being InsType of the InstructionsF of SCM st T = 0 holds JumpParts T = {0}
t4_ami_6:: for T being InsType of the InstructionsF of SCM st T = 1 holds JumpParts T = {{}}
t5_ami_6:: for T being InsType of the InstructionsF of SCM st T = 2 holds JumpParts T = {{}}
t6_ami_6:: for T being InsType of the InstructionsF of SCM st T = 3 holds JumpParts T = {{}}
t7_ami_6:: for T being InsType of the InstructionsF of SCM st T = 4 holds JumpParts T = {{}}
t8_ami_6:: for T being InsType of the InstructionsF of SCM st T = 5 holds JumpParts T = {{}}
t9_ami_6:: for T being InsType of the InstructionsF of SCM st T = 6 holds dom (product" (JumpParts T)) = {1}
d1_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for l1, l2 being Nat holds ( l1 <= l2,S iff ex f being non empty FinSequence of NAT st ( f /. 1 = l1 & f /. (len f) = l2 & ( for n being Element of NAT st 1 <= n & n < len f holds f /. (n + 1) in SUCC ((f /. n),S) ) ) );
d10_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued finite Function holds ( F is lower iff for l being Element of NAT st l in dom F holds for m being Element of NAT st m <= l,S holds m in dom F );
d11_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite Function for b4 being Element of NAT holds ( b4 = LastLoc F iff ex M being non empty finite natural-membered set st ( M = { (locnum (l,S)) where l is Element of NAT : l in dom F } & b4 = il. (S,(max M)) ) );
d12_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite Function holds ( F is halt-ending iff F . (LastLoc F) = halt S );
d13_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite Function holds ( F is unique-halt iff for f being Element of NAT st F . f = halt S & f in dom F holds f = LastLoc F );
d14_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for loc being Element of NAT for k being Nat holds loc -' (k,S) = il. (S,((locnum (loc,S)) -' k));
d2_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N holds ( S is InsLoc-antisymmetric iff for l1, l2 being Element of NAT st l1 <= l2,S & l2 <= l1,S holds l1 = l2 );
d3_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N holds ( S is weakly_standard iff ex f being Function of NAT,NAT st ( f is bijective & ( for m, n being Element of NAT holds ( m <= n iff f . m <= f . n,S ) ) ) );
d4_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for k being Nat for b4 being Element of NAT holds ( b4 = il. (S,k) iff ex f being Function of NAT,NAT st ( f is bijective & ( for m, n being Element of NAT holds ( m <= n iff f . m <= f . n,S ) ) & b4 = f . k ) );
d5_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for l, b4 being Nat holds ( b4 = locnum (l,S) iff il. (S,b4) = l );
d6_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f being Element of NAT for k being Nat holds f + (k,S) = il. (S,((locnum (f,S)) + k));
d7_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f being Element of NAT holds NextLoc (f,S) = f + (1,S);
d8_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for i being Instruction of S holds ( i is sequential iff for s being State of S holds (Exec (i,s)) . (IC ) = NextLoc ((IC s),S) );
d9_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued Function holds ( F is para-closed iff for s being State of S st IC s = il. (S,0) holds for k being Element of NAT holds IC (Comput (F,s,k)) in dom F );
t1_ami_wstd:: for l3 being Element of NAT for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for l1, l2 being Nat st l1 <= l2,S & l2 <= l3,S holds l1 <= l3,S
t10_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N holds T is InsLoc-antisymmetric
t11_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f being Element of NAT holds f + (0,T) = f by Def5;
t12_ami_wstd:: for z being Nat for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f, g being Element of NAT st f + (z,T) = g + (z,T) holds f = g
t13_ami_wstd:: for z being Nat for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f being Element of NAT holds (locnum (f,T)) + z = locnum ((f + (z,T)),T) by Def5;
t14_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f being Element of NAT holds NextLoc (f,T) = il. (T,((locnum (f,T)) + 1)) ;
t15_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f being Element of NAT holds f <> NextLoc (f,T)
t16_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for f, g being Element of NAT st NextLoc (f,T) = NextLoc (g,T) holds f = g
t17_ami_wstd:: for z being Nat for N being with_zero set holds il. ((STC N),z) = z
t18_ami_wstd:: for N being with_zero set for i being Instruction of (STC N) for s being State of (STC N) st InsCode i = 1 holds IC (Exec (i,s)) = NextLoc ((IC s),(STC N))
t19_ami_wstd:: for N being with_zero set for l being Element of NAT for i being Element of the InstructionsF of (STC N) st InsCode i = 1 holds NIC (i,l) = {(NextLoc (l,(STC N)))}
t2_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for f1, f2 being Function of NAT,NAT st f1 is bijective & ( for m, n being Element of NAT holds ( m <= n iff f1 . m <= f1 . n,S ) ) & f2 is bijective & ( for m, n being Element of NAT holds ( m <= n iff f2 . m <= f2 . n,S ) ) holds f1 = f2
t20_ami_wstd:: for N being with_zero set for l being Element of NAT holds SUCC (l,(STC N)) = {l,(NextLoc (l,(STC N)))}
t21_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for il being Element of NAT for i being Instruction of S st i is sequential holds NIC (i,il) = {(NextLoc (il,S))}
t22_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for i being Instruction of S st i is sequential holds not i is halting
t23_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued finite Function st F is really-closed & il. (S,0) in dom F holds F is para-closed
t24_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting weakly_standard AMI-Struct over N holds (il. (S,0)) .--> (halt S) is really-closed
t25_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for i being Element of the InstructionsF of T holds (il. (T,0)) .--> i is lower
t26_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite lower Function holds il. (T,0) in dom F
t27_ami_wstd:: for z being Nat for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for P being NAT -defined the InstructionsF of b3 -valued finite lower Function holds ( z < card P iff il. (T,z) in dom P )
t28_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite Function holds LastLoc F in dom F
t29_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F, G being NAT -defined the InstructionsF of b2 -valued non empty finite Function st F c= G holds LastLoc F <= LastLoc G,T
t3_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for f being Function of NAT,NAT st f is bijective holds ( ( for m, n being Element of NAT holds ( m <= n iff f . m <= f . n,S ) ) iff for k being Element of NAT holds ( f . (k + 1) in SUCC ((f . k),S) & ( for j being Element of NAT st f . j in SUCC ((f . k),S) holds k <= j ) ) )
t30_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite Function for l being Element of NAT st l in dom F holds l <= LastLoc F,T
t31_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite lower Function for G being NAT -defined the InstructionsF of b2 -valued non empty finite Function st F c= G & LastLoc F = LastLoc G holds F = G
t32_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued non empty finite lower Function holds LastLoc F = il. (T,((card F) -' 1))
t33_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for l1, l2 being Element of NAT st SUCC (l1,S) = NAT holds l1 <= l2,S
t34_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for l being Element of NAT holds l -' (0,S) = l
t35_ami_wstd:: for k being Nat for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for l being Element of NAT holds (l + (k,S)) -' (k,S) = l
t4_ami_wstd:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N holds ( S is weakly_standard iff ex f being Function of NAT,NAT st ( f is bijective & ( for k being Element of NAT holds ( f . (k + 1) in SUCC ((f . k),S) & ( for j being Element of NAT st f . j in SUCC ((f . k),S) holds k <= j ) ) ) ) )
t5_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for k1, k2 being Nat st il. (T,k1) = il. (T,k2) holds k1 = k2
t6_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for l being Nat ex k being Nat st l = il. (T,k)
t7_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for l1, l2 being Element of NAT st locnum (l1,T) = locnum (l2,T) holds l1 = l2
t8_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for k1, k2 being Nat holds ( il. (T,k1) <= il. (T,k2),T iff k1 <= k2 )
t9_ami_wstd:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated weakly_standard AMI-Struct over N for l1, l2 being Element of NAT holds ( locnum (l1,T) <= locnum (l2,T) iff l1 <= l2,T )
d1_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values AMI-Struct over N for T being InsType of the InstructionsF of S holds ( T is jump-only iff for s being State of S for o being Object of S for I being Instruction of S st InsCode I = T & o in Data-Locations holds (Exec (I,s)) . o = s . o );
d10_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued Function holds ( F is paraclosed iff for s being 0 -started State of S for P being Instruction-Sequence of S st F c= P holds for k being Element of NAT holds IC (Comput (P,s,k)) in dom F );
d11_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued Function holds ( F is parahalting iff for s being 0 -started State of S for P being Instruction-Sequence of S st F c= P holds P halts_on s );
d2_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values AMI-Struct over N for I being Instruction of S holds ( I is jump-only iff InsCode I is jump-only );
d3_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for l being Nat for i being Element of the InstructionsF of S holds NIC (i,l) = { (IC (Exec (i,ss))) where ss is Element of product (the_Values_of S) : IC ss = l } ;
d4_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for i being Element of the InstructionsF of S holds JUMP i = meet { (NIC (i,l)) where l is Element of NAT : verum } ;
d5_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for l being Nat holds SUCC (l,S) = union { ((NIC (i,l)) \ (JUMP i)) where i is Element of the InstructionsF of S : verum } ;
d6_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N holds ( S is standard iff for m, n being Element of NAT holds ( m <= n iff ex f being non empty FinSequence of NAT st ( f /. 1 = m & f /. (len f) = n & ( for n being Element of NAT st 1 <= n & n < len f holds f /. (n + 1) in SUCC ((f /. n),S) ) ) ) );
d7_amistd_1:: for N being with_zero set for b2 being strict AMI-Struct over N holds ( b2 = STC N iff ( the carrier of b2 = {0} & the ZeroF of b2 = 0 & the InstructionsF of b2 = {[0,0,0],[1,0,0]} & the Object-Kind of b2 = 0 .--> 0 & the ValuesF of b2 = N --> NAT & ex f being Function of (product (the_Values_of b2)),(product (the_Values_of b2)) st ( ( for s being Element of product (the_Values_of b2) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of b2 = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product (the_Values_of b2)))) ) ) );
d8_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for i being Instruction of S holds ( i is sequential iff for s being State of S holds (Exec (i,s)) . (IC ) = succ (IC s) );
d9_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for F being preProgram of S holds ( F is really-closed iff for l being Element of NAT st l in dom F holds NIC ((F /. l),l) c= dom F );
t1_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for i being Element of the InstructionsF of S st ( for l being Element of NAT holds NIC (i,l) = {l} ) holds JUMP i is empty
t10_amistd_1:: for N being with_zero set for l being Element of NAT for i being Element of the InstructionsF of (STC N) st InsCode i = 1 holds NIC (i,l) = {(succ l)} by Lm4;
t11_amistd_1:: for N being with_zero set for l being Element of NAT holds SUCC (l,(STC N)) = {l,(succ l)} by Th8;
t12_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for il being Element of NAT for i being Instruction of S st i is sequential holds NIC (i,il) = {(succ il)}
t13_amistd_1:: for N being with_zero set for T being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for i being Instruction of T st not JUMP i is empty holds not i is sequential
t14_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for F being preProgram of S holds ( F is really-closed iff for s being State of S st IC s in dom F holds for k being Element of NAT holds IC (Comput (F,s,k)) in dom F )
t15_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for F being NAT -defined the InstructionsF of b2 -valued finite Function st F is really-closed & 0 in dom F holds F is paraclosed
t16_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting standard AMI-Struct over N holds 0 .--> (halt S) is really-closed
t17_amistd_1:: for N being with_zero set for i being Instruction of (Trivial-AMI N) holds i is halting
t18_amistd_1:: for N being with_zero set for i being Element of the InstructionsF of (Trivial-AMI N) holds InsCode i = 0
t19_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for i being Instruction of S for l being Element of NAT holds JUMP i c= NIC (i,l)
t2_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for il being Element of NAT for i being Instruction of S st i is halting holds NIC (i,il) = {il}
t20_amistd_1:: for N being with_zero set for i being Instruction of (STC N) for s being State of (STC N) st InsCode i = 1 holds Exec (i,s) = IncIC (s,1)
t21_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for F being preProgram of S holds ( F is really-closed iff for s being State of S st IC s in dom F holds for P being Instruction-Sequence of S st F c= P holds for k being Element of NAT holds IC (Comput (P,s,k)) in dom F ) by Lm6;
t3_amistd_1:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N holds ( S is standard iff for k being Element of NAT holds ( k + 1 in SUCC (k,S) & ( for j being Element of NAT st j in SUCC (k,S) holds k <= j ) ) )
t4_amistd_1:: for N being with_zero set for i being Instruction of (STC N) st InsCode i = 0 holds i is halting
t5_amistd_1:: for N being with_zero set for i being Instruction of (STC N) st InsCode i = 1 holds not i is halting
t6_amistd_1:: for N being with_zero set for i being Element of the InstructionsF of (STC N) holds ( InsCode i = 1 or InsCode i = 0 )
t7_amistd_1:: for N being with_zero set for i being Instruction of (STC N) holds i is jump-only
t8_amistd_1:: for z being Nat for N being with_zero set for l being Element of NAT st l = z holds SUCC (l,(STC N)) = {z,(z + 1)}
t9_amistd_1:: for N being with_zero set for i being Instruction of (STC N) for s being State of (STC N) st InsCode i = 1 holds (Exec (i,s)) . (IC ) = succ (IC s) by Lm3;
d1_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of S holds ( I is with_explicit_jumps iff JUMP I = rng (JumpPart I) );
d2_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N holds ( S is with_explicit_jumps iff for I being Instruction of S holds I is with_explicit_jumps );
d3_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N for I being Instruction of S holds ( I is IC-relocable iff for j, k being Nat for s being State of S holds (IC (Exec ((IncAddr (I,j)),s))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s,k)))) );
d4_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N holds ( S is IC-relocable iff for I being Instruction of S holds I is IC-relocable );
t1_amistd_2:: for N being with_zero set for I being Instruction of (STC N) holds JumpPart I = 0 ;
t10_amistd_2:: for N being with_zero set for n being Element of NAT for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for s being State of S for I being Program of for P1, P2 being Instruction-Sequence of S st I c= P1 & I c= P2 & ( for m being Element of NAT st m < n holds IC (Comput (P2,s,m)) in dom I ) holds for m being Element of NAT st m <= n holds Comput (P1,s,m) = Comput (P2,s,m)
t11_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N for P being Instruction-Sequence of S for s being State of S st s = Following (P,s) holds for n being Element of NAT holds Comput (P,s,n) = s
t2_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for I being Instruction of S st ( for f being Element of NAT holds NIC (I,f) = {(succ f)} ) holds JUMP I is empty
t3_amistd_2:: for N being with_zero set for T being InsType of the InstructionsF of (STC N) holds JumpParts T = {0}
t4_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of S st I is halting holds JUMP I is empty
t5_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N for I being Instruction of S st I is ins-loc-free holds JUMP I is empty
t6_amistd_2:: for k being Nat for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting standard with_explicit_jumps AMI-Struct over N for I being Instruction of S st I is sequential holds IncAddr (I,k) is sequential by COMPOS_0:4;
t7_amistd_2:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting with_explicit_jumps AMI-Struct over N for I being IC-relocable Instruction of S for k being Nat for s being State of S holds (IC (Exec (I,s))) + k = IC (Exec ((IncAddr (I,k)),(IncIC (s,k))))
t8_amistd_2:: for N being with_zero set for I being Instruction of (Trivial-AMI N) holds JumpPart I = 0 by Lm1;
t9_amistd_2:: for N being with_zero set for T being InsType of the InstructionsF of (Trivial-AMI N) holds JumpParts T = {0} by Lm2;
d1_amistd_3:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for M being Subset of NAT for b4 being T-Sequence of NAT holds ( b4 = LocSeq (M,S) iff ( dom b4 = card M & ( for m being set st m in card M holds b4 . m = (canonical_isomorphism_of ((RelIncl (order_type_of (RelIncl M))),(RelIncl M))) . m ) ) );
d2_amistd_3:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for M being non empty preProgram of S for b4 being DecoratedTree of NAT holds ( b4 = ExecTree M iff ( b4 . {} = FirstLoc M & ( for t being Element of dom b4 holds ( succ t = { (t ^ <*k*>) where k is Element of NAT : k in card (NIC ((M /. (b4 . t)),(b4 . t))) } & ( for m being Element of NAT st m in card (NIC ((M /. (b4 . t)),(b4 . t))) holds b4 . (t ^ <*m*>) = (LocSeq ((NIC ((M /. (b4 . t)),(b4 . t))),S)) . m ) ) ) ) );
t1_amistd_3:: for n being Nat for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for F being Subset of NAT st F = {n} holds LocSeq (F,S) = 0 .--> n
t2_amistd_3:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting standard AMI-Struct over N holds ExecTree (Stop S) = TrivialInfiniteTree --> 0
d1_amistd_4:: for A being COM-Struct holds ( A is with_non_trivial_Instructions iff not the InstructionsF of A is trivial );
d2_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values AMI-Struct over N holds ( A is with_non_trivial_ObjectKinds iff for o being Object of A holds not Values o is trivial );
d3_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values AMI-Struct over N for I being Instruction of A for b4 being Subset of A holds ( b4 = Output I iff for o being Object of A holds ( o in b4 iff ex s being State of A st s . o <> (Exec (I,s)) . o ) );
d4_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A for b4 being Subset of A holds ( b4 = Out_\_Inp I iff for o being Object of A holds ( o in b4 iff for s being State of A for a being Element of Values o holds Exec (I,s) = Exec (I,(s +* (o,a))) ) );
d5_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A for b4 being Subset of A holds ( b4 = Out_U_Inp I iff for o being Object of A holds ( o in b4 iff ex s being State of A ex a being Element of Values o st Exec (I,(s +* (o,a))) <> (Exec (I,s)) +* (o,a) ) );
d6_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A holds Input I = (Out_U_Inp I) \ (Out_\_Inp I);
t1_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for I being Instruction of A for s being State of A for o being Object of A for w being Element of Values o st I is sequential & o <> IC holds IC (Exec (I,s)) = IC (Exec (I,(s +* (o,w))))
t10_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A holds ( I is halting iff Output I is empty )
t11_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated with_non_trivial_ObjectKinds AMI-Struct over N for I being Instruction of A st I is halting holds Out_\_Inp I is empty
t12_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A st I is halting holds Out_U_Inp I is empty
t13_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A st I is halting holds Input I is empty
t14_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for I being Instruction of A st I is sequential holds not IC in Out_\_Inp I
t15_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A st ex s being State of A st (Exec (I,s)) . (IC ) <> IC s holds IC in Output I by Def3;
t16_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for I being Instruction of A st I is sequential holds IC in Output I
t17_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A st ex s being State of A st (Exec (I,s)) . (IC ) <> IC s holds IC in Out_U_Inp I
t18_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for I being Instruction of A st I is sequential holds IC in Out_U_Inp I
t19_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A for o being Object of A st I is jump-only & o in Output I holds o = IC
t2_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated standard AMI-Struct over N for I being Instruction of A for s being State of A for o being Object of A for w being Element of Values o st I is sequential & o <> IC holds IC (Exec (I,(s +* (o,w)))) = IC ((Exec (I,s)) +* (o,w))
t3_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated with_non_trivial_ObjectKinds AMI-Struct over N for I being Instruction of A holds Out_\_Inp I c= Output I
t4_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A holds Output I c= Out_U_Inp I
t5_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated with_non_trivial_ObjectKinds AMI-Struct over N for I being Instruction of A holds Out_\_Inp I = (Output I) \ (Input I)
t6_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated with_non_trivial_ObjectKinds AMI-Struct over N for I being Instruction of A holds Out_U_Inp I = (Output I) \/ (Input I)
t7_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A for o being Object of A st Values o is trivial holds not o in Out_U_Inp I
t8_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A for o being Object of A st Values o is trivial holds not o in Input I
t9_amistd_4:: for N being with_zero set for A being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of A for o being Object of A st Values o is trivial holds not o in Output I
d1_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N for I being Instruction of S holds ( I is relocable iff for j, k being Nat for s being State of S holds Exec ((IncAddr (I,(j + k))),(IncIC (s,k))) = IncIC ((Exec ((IncAddr (I,j)),s)),k) );
d2_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N holds ( S is relocable iff for I being Instruction of S holds I is relocable );
d3_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N holds ( S is IC-recognized iff for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being b3 -autonomic FinPartState of S st not p is empty holds IC in dom p );
d4_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N holds ( S is CurIns-recognized iff for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b3 -autonomic FinPartState of S for s being State of S st p c= s holds for P being Instruction-Sequence of S st q c= P holds for i being Element of NAT holds IC (Comput (P,s,i)) in dom q );
d5_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized AMI-Struct over N holds ( S is relocable1 iff for k being Element of NAT for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b4 -autonomic FinPartState of S for s1, s2 being State of S st p c= s1 & IncIC (p,k) c= s2 holds for P1, P2 being Instruction-Sequence of S st q c= P1 & Reloc (q,k) c= P2 holds for i being Element of NAT holds IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) );
d6_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized AMI-Struct over N holds ( S is relocable2 iff for k being Element of NAT for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b4 -autonomic FinPartState of S for s1, s2 being State of S st p c= s1 & IncIC (p,k) c= s2 holds for P1, P2 being Instruction-Sequence of S st q c= P1 & Reloc (q,k) c= P2 holds for i being Element of NAT holds (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) );
d7_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N for k being Nat for F being NAT -defined the InstructionsF of b2 -valued Function holds ( F is k -halting iff for s being b3 -started State of S for P being Instruction-Sequence of S st F c= P holds P halts_on s );
t1_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N for I being Instruction of S st I is jump-only holds for k being Element of NAT holds IncAddr (I,k) is jump-only
t10_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized AMI-Struct over N for k being Element of NAT for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty FinPartState of S st IC in dom p holds for s being State of S st p c= s & IncIC (p,k) is Reloc (q,k) -autonomic holds for P being Instruction-Sequence of S st q c= P holds for i being Element of NAT holds Comput (P,s,i) = DecIC ((Comput ((P +* (Reloc (q,k))),(s +* (IncIC (p,k))),i)),k)
t11_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized AMI-Struct over N for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty FinPartState of S st IC in dom p holds for k being Element of NAT holds ( p is q -autonomic iff IncIC (p,k) is Reloc (q,k) -autonomic )
t12_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized relocable1 relocable2 AMI-Struct over N for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b3 -autonomic FinPartState of S for k being Element of NAT st IC in dom p holds ( p is q -halted iff IncIC (p,k) is Reloc (q,k) -halted )
t13_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized relocable1 relocable2 AMI-Struct over N for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b3 -autonomic b3 -halted FinPartState of S st IC in dom p holds for k being Element of NAT holds DataPart (Result (q,p)) = DataPart (Result ((Reloc (q,k)),(IncIC (p,k))))
t14_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized relocable1 relocable2 AMI-Struct over N for F being data-only PartFunc of (FinPartSt S),(FinPartSt S) for l being Element of NAT for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b5 -autonomic b5 -halted FinPartState of S st IC in dom p holds for k being Element of NAT holds ( q,p computes F iff Reloc (q,k), IncIC (p,k) computes F )
t15_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting IC-recognized CurIns-recognized AMI-Struct over N for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being b3 -autonomic FinPartState of S st IC in dom p holds IC p in dom q
t2_amistd_5:: for N being with_zero set for I being Instruction of (STC N) for s being State of (STC N) for k being Nat holds Exec (I,(IncIC (s,k))) = IncIC ((Exec (I,s)),k)
t3_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N holds ( S is IC-recognized iff for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being b3 -autonomic FinPartState of S st DataPart p <> {} holds IC in dom p )
t4_amistd_5:: for k being Nat for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable AMI-Struct over N for INS being Instruction of S for s being State of S holds Exec ((IncAddr (INS,k)),(IncIC (s,k))) = IncIC ((Exec (INS,s)),k)
t5_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable AMI-Struct over N for INS being Instruction of S for s being State of S for j, k being Element of NAT st IC s = j + k holds Exec (INS,(DecIC (s,k))) = DecIC ((Exec ((IncAddr (INS,k)),s)),k)
t6_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting IC-recognized AMI-Struct over N for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b3 -autonomic FinPartState of S holds IC in dom p
t7_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting IC-recognized CurIns-recognized AMI-Struct over N for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being non empty b3 -autonomic FinPartState of S for s1, s2 being State of S st p c= s1 & p c= s2 holds for P1, P2 being Instruction-Sequence of S st q c= P1 & q c= P2 holds for i being Element of NAT holds ( IC (Comput (P1,s1,i)) = IC (Comput (P2,s2,i)) & CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) )
t8_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized AMI-Struct over N for k being Element of NAT for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being b4 -autonomic FinPartState of S st IC in dom p holds for s being State of S st p c= s holds for P being Instruction-Sequence of S st q c= P holds for i being Element of NAT holds Comput ((P +* (Reloc (q,k))),(s +* (IncIC (p,k))),i) = IncIC ((Comput (P,s,i)),k)
t9_amistd_5:: for N being with_zero set for S being non empty with_non-empty_values IC-Ins-separated halting relocable IC-recognized CurIns-recognized AMI-Struct over N for k being Element of NAT for q being NAT -defined the InstructionsF of b2 -valued finite non halt-free Function for p being b4 -autonomic FinPartState of S st IC in dom p holds for s being State of S st IncIC (p,k) c= s holds for P being Instruction-Sequence of S st Reloc (q,k) c= P holds for i being Element of NAT holds Comput (P,s,i) = IncIC ((Comput ((P +* q),(s +* p),i)),k)
d1_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V holds ( Gen w,y iff ( ( for u being VECTOR of V ex a1, a2 being Real st u = (a1 * w) + (a2 * y) ) & ( for a1, a2 being Real st (a1 * w) + (a2 * y) = 0. V holds ( a1 = 0 & a2 = 0 ) ) ) );
d10_analmetr:: for POS being non empty ParOrtStr for a, b, c being Element of POS holds ( LIN a,b,c iff a,b // a,c );
d11_analmetr:: for POS being non empty ParOrtStr for a, b being Element of POS for b4 being Subset of POS holds ( b4 = Line (a,b) iff for x being Element of POS holds ( x in b4 iff LIN a,b,x ) );
d12_analmetr:: for POS being non empty ParOrtStr for A being Subset of POS holds ( A is being_line iff ex a, b being Element of POS st ( a <> b & A = Line (a,b) ) );
d13_analmetr:: for POS being non empty ParOrtStr for a, b being Element of POS for K being Subset of POS holds ( a,b _|_ K iff ex p, q being Element of POS st ( p <> q & K = Line (p,q) & a,b _|_ p,q ) );
d14_analmetr:: for POS being non empty ParOrtStr for K, M being Subset of POS holds ( K _|_ M iff ex p, q being Element of POS st ( p <> q & K = Line (p,q) & p,q _|_ M ) );
d15_analmetr:: for POS being non empty ParOrtStr for K, M being Subset of POS holds ( K // M iff ex a, b, c, d being Element of POS st ( a <> b & c <> d & K = Line (a,b) & M = Line (c,d) & a,b // c,d ) );
d2_analmetr:: for V being RealLinearSpace for u, v, w, y being VECTOR of V holds ( u,v are_Ort_wrt w,y iff ex a1, a2, b1, b2 being Real st ( u = (a1 * w) + (a2 * y) & v = (b1 * w) + (b2 * y) & (a1 * b1) + (a2 * b2) = 0 ) );
d3_analmetr:: for V being RealLinearSpace for u, u1, v, v1, w, y being VECTOR of V holds ( u,u1,v,v1 are_Ort_wrt w,y iff u1 - u,v1 - v are_Ort_wrt w,y );
d4_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for b4 being Relation of [: the carrier of V, the carrier of V:] holds ( b4 = Orthogonality (V,w,y) iff for x, z being set holds ( [x,z] in b4 iff ex u, u1, v, v1 being VECTOR of V st ( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_Ort_wrt w,y ) ) );
d5_analmetr:: for POS being non empty ParOrtStr for a, b, c, d being Element of POS holds ( a,b _|_ c,d iff [[a,b],[c,d]] in the orthogonality of POS );
d6_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V holds AMSpace (V,w,y) = ParOrtStr(# the carrier of V,(lambda (DirPar V)),(Orthogonality (V,w,y)) #);
d7_analmetr:: for POS being non empty ParOrtStr holds Af POS = AffinStruct(# the carrier of POS, the CONGR of POS #);
d8_analmetr:: for IT being non empty ParOrtStr holds ( IT is OrtAfSp-like iff ( AffinStruct(# the carrier of IT, the CONGR of IT #) is AffinSpace & ( for a, b, c, d, p, q, r, s being Element of IT holds ( ( a,b _|_ a,b implies a = b ) & a,b _|_ c,c & ( a,b _|_ c,d implies ( a,b _|_ d,c & c,d _|_ a,b ) ) & ( a,b _|_ p,q & a,b // r,s & not p,q _|_ r,s implies a = b ) & ( a,b _|_ p,q & a,b _|_ p,s implies a,b _|_ q,s ) ) ) & ( for a, b, c being Element of IT st a <> b holds ex x being Element of IT st ( a,b // a,x & a,b _|_ x,c ) ) & ( for a, b, c being Element of IT ex x being Element of IT st ( a,b _|_ c,x & c <> x ) ) ) );
d9_analmetr:: for IT being non empty ParOrtStr holds ( IT is OrtAfPl-like iff ( AffinStruct(# the carrier of IT, the CONGR of IT #) is AffinPlane & ( for a, b, c, d, p, q, r, s being Element of IT holds ( ( a,b _|_ a,b implies a = b ) & a,b _|_ c,c & ( a,b _|_ c,d implies ( a,b _|_ d,c & c,d _|_ a,b ) ) & ( a,b _|_ p,q & a,b // r,s & not p,q _|_ r,s implies a = b ) & ( a,b _|_ p,q & a,b _|_ r,s & not p,q // r,s implies a = b ) ) ) & ( for a, b, c being Element of IT ex x being Element of IT st ( a,b _|_ c,x & c <> x ) ) ) );
t1_analmetr:: for V being RealLinearSpace for u, v, w, y being VECTOR of V st Gen w,y holds ( u,v are_Ort_wrt w,y iff for a1, a2, b1, b2 being Real st u = (a1 * w) + (a2 * y) & v = (b1 * w) + (b2 * y) holds (a1 * b1) + (a2 * b2) = 0 )
t10_analmetr:: for V being RealLinearSpace for w, y, u, v1, v2 being VECTOR of V st Gen w,y & u,v1 are_Ort_wrt w,y & u,v2 are_Ort_wrt w,y holds ( u,v1 + v2 are_Ort_wrt w,y & u,v1 - v2 are_Ort_wrt w,y )
t11_analmetr:: for V being RealLinearSpace for w, y, u being VECTOR of V st Gen w,y & u,u are_Ort_wrt w,y holds u = 0. V
t12_analmetr:: for V being RealLinearSpace for w, y, u, u1, u2 being VECTOR of V st Gen w,y & u,u1 - u2 are_Ort_wrt w,y & u1,u2 - u are_Ort_wrt w,y holds u2,u - u1 are_Ort_wrt w,y
t13_analmetr:: for V being RealLinearSpace for w, y, u, v being VECTOR of V st Gen w,y & u <> 0. V holds ex a being Real st v - (a * u),u are_Ort_wrt w,y
t14_analmetr:: for V being RealLinearSpace for u, v, u1, v1 being VECTOR of V holds ( ( u,v // u1,v1 or u,v // v1,u1 ) iff ex a, b being Real st ( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) )
t15_analmetr:: for V being RealLinearSpace for u, v, u1, v1 being VECTOR of V holds ( [[u,v],[u1,v1]] in lambda (DirPar V) iff ex a, b being Real st ( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) )
t16_analmetr:: for V being RealLinearSpace holds the carrier of (Lambda (OASpace V)) = the carrier of V
t17_analmetr:: for V being RealLinearSpace holds the CONGR of (Lambda (OASpace V)) = lambda (DirPar V)
t18_analmetr:: for V being RealLinearSpace for u, v, u1, v1 being VECTOR of V for p, q, p1, q1 being Element of (Lambda (OASpace V)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff ex a, b being Real st ( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) )
t19_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V holds ( the carrier of (AMSpace (V,w,y)) = the carrier of V & the CONGR of (AMSpace (V,w,y)) = lambda (DirPar V) & the orthogonality of (AMSpace (V,w,y)) = Orthogonality (V,w,y) ) ;
t2_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V holds w,y are_Ort_wrt w,y
t20_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V holds Af (AMSpace (V,w,y)) = Lambda (OASpace V)
t21_analmetr:: for V being RealLinearSpace for u, u1, v, v1, w, y being VECTOR of V for p, p1, q, q1 being Element of (AMSpace (V,w,y)) st p = u & p1 = u1 & q = v & q1 = v1 holds ( p,q _|_ p1,q1 iff u,v,u1,v1 are_Ort_wrt w,y )
t22_analmetr:: for V being RealLinearSpace for w, y, u, v, u1, v1 being VECTOR of V for p, q, p1, q1 being Element of (AMSpace (V,w,y)) st p = u & q = v & p1 = u1 & q1 = v1 holds ( p,q // p1,q1 iff ex a, b being Real st ( a * (v - u) = b * (v1 - u1) & ( a <> 0 or b <> 0 ) ) )
t23_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, q, p1, q1 being Element of (AMSpace (V,w,y)) st p,q _|_ p1,q1 holds p1,q1 _|_ p,q
t24_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, q, p1, q1 being Element of (AMSpace (V,w,y)) st p,q _|_ p1,q1 holds p,q _|_ q1,p1
t25_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for p, q, r being Element of (AMSpace (V,w,y)) holds p,q _|_ r,r
t26_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, p1, q, q1, r, r1 being Element of (AMSpace (V,w,y)) st p,p1 _|_ q,q1 & p,p1 // r,r1 & not p = p1 holds q,q1 _|_ r,r1
t27_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for p, q, r being Element of (AMSpace (V,w,y)) ex r1 being Element of (AMSpace (V,w,y)) st ( p,q _|_ r,r1 & r <> r1 )
t28_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, p1, q, q1, r, r1 being Element of (AMSpace (V,w,y)) st Gen w,y & p,p1 _|_ q,q1 & p,p1 _|_ r,r1 & not p = p1 holds q,q1 // r,r1
t29_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, q, r, r1, r2 being Element of (AMSpace (V,w,y)) st Gen w,y & p,q _|_ r,r1 & p,q _|_ r,r2 holds p,q _|_ r1,r2
t3_analmetr:: ex V being RealLinearSpace ex w, y being VECTOR of V st Gen w,y
t30_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, q being Element of (AMSpace (V,w,y)) st Gen w,y & p,q _|_ p,q holds p = q
t31_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, q, p1, p2 being Element of (AMSpace (V,w,y)) st Gen w,y & p,q _|_ p1,p2 & p1,q _|_ p2,p holds p2,q _|_ p,p1
t32_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V for p, p1 being Element of (AMSpace (V,w,y)) st Gen w,y & p <> p1 holds for q being Element of (AMSpace (V,w,y)) ex q1 being Element of (AMSpace (V,w,y)) st ( p,p1 // p,q1 & p,p1 _|_ q1,q )
t33_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds AMSpace (V,w,y) is OrtAfSp
t34_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds AMSpace (V,w,y) is OrtAfPl
t35_analmetr:: for POS being non empty ParOrtStr for x being set holds ( x is Element of POS iff x is Element of (Af POS) ) ;
t36_analmetr:: for POS being non empty ParOrtStr for a, b, c, d being Element of POS for a9, b9, c9, d9 being Element of (Af POS) st a = a9 & b = b9 & c = c9 & d = d9 holds ( a,b // c,d iff a9,b9 // c9,d9 )
t37_analmetr:: for POS being OrtAfPl holds POS is OrtAfSp
t38_analmetr:: for POS being OrtAfSp st Af POS is AffinPlane holds POS is OrtAfPl
t39_analmetr:: for POS being non empty ParOrtStr holds ( POS is OrtAfPl-like iff ( ex a, b being Element of POS st a <> b & ( for a, b, c, d, p, q, r, s being Element of POS holds ( a,b // b,a & a,b // c,c & ( a,b // p,q & a,b // r,s & not p,q // r,s implies a = b ) & ( a,b // a,c implies b,a // b,c ) & ex x being Element of POS st ( a,b // c,x & a,c // b,x ) & not for x, y, z being Element of POS holds x,y // x,z & ex x being Element of POS st ( a,b // c,x & c <> x ) & ( a,b // b,d & b <> a implies ex x being Element of POS st ( c,b // b,x & c,a // d,x ) ) & ( a,b _|_ a,b implies a = b ) & a,b _|_ c,c & ( a,b _|_ c,d implies ( a,b _|_ d,c & c,d _|_ a,b ) ) & ( a,b _|_ p,q & a,b // r,s & not p,q _|_ r,s implies a = b ) & ( a,b _|_ p,q & a,b _|_ r,s & not p,q // r,s implies a = b ) & ex x being Element of POS st ( a,b _|_ c,x & c <> x ) & ( not a,b // c,d implies ex x being Element of POS st ( a,b // a,x & c,d // c,x ) ) ) ) ) )
t4_analmetr:: for V being RealLinearSpace for u, v, w, y being VECTOR of V st u,v are_Ort_wrt w,y holds v,u are_Ort_wrt w,y
t40_analmetr:: for POS being OrtAfSp for a, b, c being Element of POS for a9, b9, c9 being Element of (Af POS) st a = a9 & b = b9 & c = c9 holds ( LIN a,b,c iff LIN a9,b9,c9 )
t41_analmetr:: for POS being OrtAfSp for a, b being Element of POS for a9, b9 being Element of (Af POS) st a = a9 & b = b9 holds Line (a,b) = Line (a9,b9)
t42_analmetr:: for POS being non empty ParOrtStr for X being set holds ( X is Subset of POS iff X is Subset of (Af POS) ) ;
t43_analmetr:: for POS being OrtAfSp for X being Subset of POS for Y being Subset of (Af POS) st X = Y holds ( X is being_line iff Y is being_line )
t44_analmetr:: for POS being non empty ParOrtStr for a, b being Element of POS for K, M being Subset of POS holds ( ( a,b _|_ K implies K is being_line ) & ( K _|_ M implies ( K is being_line & M is being_line ) ) )
t45_analmetr:: for POS being non empty ParOrtStr for K, M being Subset of POS holds ( K _|_ M iff ex a, b, c, d being Element of POS st ( a <> b & c <> d & K = Line (a,b) & M = Line (c,d) & a,b _|_ c,d ) )
t46_analmetr:: for POS being OrtAfSp for M, N being Subset of POS for M9, N9 being Subset of (Af POS) st M = M9 & N = N9 holds ( M // N iff M9 // N9 )
t47_analmetr:: for POS being OrtAfSp for K being Subset of POS for a being Element of POS st K is being_line holds a,a _|_ K
t48_analmetr:: for POS being OrtAfSp for K being Subset of POS for a, b, c, d being Element of POS st a,b _|_ K & ( a,b // c,d or c,d // a,b ) & a <> b holds c,d _|_ K
t49_analmetr:: for POS being OrtAfSp for K being Subset of POS for a, b being Element of POS st a,b _|_ K holds b,a _|_ K
t5_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u, v being VECTOR of V holds ( u, 0. V are_Ort_wrt w,y & 0. V,v are_Ort_wrt w,y )
t50_analmetr:: for POS being OrtAfSp for K, M being Subset of POS for r, s being Element of POS st r,s _|_ K & K // M holds r,s _|_ M
t51_analmetr:: for POS being OrtAfSp for K being Subset of POS for a, b being Element of POS st a in K & b in K & a,b _|_ K holds a = b
t52_analmetr:: for POS being OrtAfSp for K, M, N being Subset of POS st K _|_ M & K // N holds N _|_ M
t53_analmetr:: for POS being OrtAfSp for K being Subset of POS for a, b, c, d being Element of POS st a in K & b in K & c,d _|_ K holds ( c,d _|_ a,b & a,b _|_ c,d )
t54_analmetr:: for POS being OrtAfSp for K being Subset of POS for a, b being Element of POS st a in K & b in K & a <> b & K is being_line holds K = Line (a,b)
t55_analmetr:: for POS being OrtAfSp for K being Subset of POS for a, b, c, d being Element of POS st a in K & b in K & a <> b & K is being_line & ( a,b _|_ c,d or c,d _|_ a,b ) holds c,d _|_ K
t56_analmetr:: for POS being OrtAfSp for M, N being Subset of POS for a, b, c, d being Element of POS st a in M & b in M & c in N & d in N & M _|_ N holds a,b _|_ c,d
t57_analmetr:: for POS being OrtAfSp for M, N, K, A being Subset of POS for p, a, b being Element of POS st p in M & p in N & a in M & b in N & a <> b & a in K & b in K & A _|_ M & A _|_ N & K is being_line holds A _|_ K
t58_analmetr:: for POS being OrtAfSp for b, c, a being Element of POS holds ( b,c _|_ a,a & a,a _|_ b,c & b,c // a,a & a,a // b,c )
t59_analmetr:: for POS being OrtAfSp for a, b, c, d being Element of POS st a,b // c,d holds ( a,b // d,c & b,a // c,d & b,a // d,c & c,d // a,b & c,d // b,a & d,c // a,b & d,c // b,a )
t6_analmetr:: for V being RealLinearSpace for u, v, w, y being VECTOR of V for a, b being Real st u,v are_Ort_wrt w,y holds a * u,b * v are_Ort_wrt w,y
t60_analmetr:: for POS being OrtAfSp for p, q, a, b, c, d being Element of POS st p <> q & ( ( p,q // a,b & p,q // c,d ) or ( p,q // a,b & c,d // p,q ) or ( a,b // p,q & c,d // p,q ) or ( a,b // p,q & p,q // c,d ) ) holds a,b // c,d
t61_analmetr:: for POS being OrtAfSp for a, b, c, d being Element of POS st a,b _|_ c,d holds ( a,b _|_ d,c & b,a _|_ c,d & b,a _|_ d,c & c,d _|_ a,b & c,d _|_ b,a & d,c _|_ a,b & d,c _|_ b,a )
t62_analmetr:: for POS being OrtAfSp for p, q, a, b, c, d being Element of POS st p <> q & ( ( p,q // a,b & p,q _|_ c,d ) or ( p,q // c,d & p,q _|_ a,b ) or ( p,q // a,b & c,d _|_ p,q ) or ( p,q // c,d & a,b _|_ p,q ) or ( a,b // p,q & c,d _|_ p,q ) or ( c,d // p,q & a,b _|_ p,q ) or ( a,b // p,q & p,q _|_ c,d ) or ( c,d // p,q & p,q _|_ a,b ) ) holds a,b _|_ c,d
t63_analmetr:: for POS being OrtAfPl for p, q, a, b, c, d being Element of POS st p <> q & ( ( p,q _|_ a,b & p,q _|_ c,d ) or ( p,q _|_ a,b & c,d _|_ p,q ) or ( a,b _|_ p,q & c,d _|_ p,q ) or ( a,b _|_ p,q & p,q _|_ c,d ) ) holds a,b // c,d
t64_analmetr:: for POS being OrtAfPl for M, N being Subset of POS for a, b, c, d being Element of POS st a in M & b in M & a <> b & M is being_line & c in N & d in N & c <> d & N is being_line & a,b // c,d holds M // N
t65_analmetr:: for POS being OrtAfPl for M, K, N being Subset of POS st M _|_ K & N _|_ K holds M // N
t66_analmetr:: for POS being OrtAfPl for M, N being Subset of POS st M _|_ N holds ex p being Element of POS st ( p in M & p in N )
t67_analmetr:: for POS being OrtAfPl for a, b, c, d being Element of POS st a,b _|_ c,d holds ex p being Element of POS st ( LIN a,b,p & LIN c,d,p )
t68_analmetr:: for POS being OrtAfPl for K being Subset of POS for a, b being Element of POS st a,b _|_ K holds ex p being Element of POS st ( LIN a,b,p & p in K )
t69_analmetr:: for POS being OrtAfPl for a, p, q being Element of POS ex x being Element of POS st ( a,x _|_ p,q & LIN p,q,x )
t7_analmetr:: for V being RealLinearSpace for u, v, w, y being VECTOR of V for a, b being Real st u,v are_Ort_wrt w,y holds ( a * u,v are_Ort_wrt w,y & u,b * v are_Ort_wrt w,y )
t70_analmetr:: for POS being OrtAfPl for K being Subset of POS for a being Element of POS st K is being_line holds ex x being Element of POS st ( a,x _|_ K & x in K )
t8_analmetr:: for V being RealLinearSpace for w, y being VECTOR of V st Gen w,y holds for u being VECTOR of V ex v being VECTOR of V st ( u,v are_Ort_wrt w,y & v <> 0. V )
t9_analmetr:: for V being RealLinearSpace for w, y, v, u1, u2 being VECTOR of V st Gen w,y & v,u1 are_Ort_wrt w,y & v,u2 are_Ort_wrt w,y & v <> 0. V holds ex a, b being Real st ( a * u1 = b * u2 & ( a <> 0 or b <> 0 ) )
d1_analoaf:: for V being RealLinearSpace for u, v, w, y being VECTOR of V holds ( u,v // w,y iff ( u = v or w = y or ex a, b being Real st ( 0 < a & 0 < b & a * (v - u) = b * (y - w) ) ) );
d2_analoaf:: for AS being non empty AffinStruct for a, b, c, d being Element of AS holds ( a,b // c,d iff [[a,b],[c,d]] in the CONGR of AS );
d3_analoaf:: for V being RealLinearSpace for b2 being Relation of [: the carrier of V, the carrier of V:] holds ( b2 = DirPar V iff for x, z being set holds ( [x,z] in b2 iff ex u, v, w, y being VECTOR of V st ( x = [u,v] & z = [w,y] & u,v // w,y ) ) );
d4_analoaf:: for V being RealLinearSpace holds OASpace V = AffinStruct(# the carrier of V,(DirPar V) #);
d5_analoaf:: for IT being non empty AffinStruct holds ( IT is OAffinSpace-like iff ( ( for a, b, c, d, p, q, r, s being Element of IT holds ( a,b // c,c & ( a,b // b,a implies a = b ) & ( a <> b & a,b // p,q & a,b // r,s implies p,q // r,s ) & ( a,b // c,d implies b,a // d,c ) & ( a,b // b,c implies a,b // a,c ) & ( not a,b // a,c or a,b // b,c or a,c // c,b ) ) ) & ex a, b, c, d being Element of IT st ( not a,b // c,d & not a,b // d,c ) & ( for a, b, c being Element of IT ex d being Element of IT st ( a,b // c,d & a,c // b,d & b <> d ) ) & ( for p, a, b, c being Element of IT st p <> b & b,p // p,c holds ex d being Element of IT st ( a,p // p,d & a,b // c,d ) ) ) );
d6_analoaf:: for IT being OAffinSpace holds ( IT is 2-dimensional iff for a, b, c, d being Element of IT st not a,b // c,d & not a,b // d,c holds ex p being Element of IT st ( ( a,b // a,p or a,b // p,a ) & ( c,d // c,p or c,d // p,c ) ) );
t1_analoaf:: for V being RealLinearSpace for w, v, u being VECTOR of V holds (w - v) + (v - u) = w - u
t10_analoaf:: for V being RealLinearSpace for u, v being VECTOR of V st u,v // v,u holds u = v
t11_analoaf:: for V being RealLinearSpace for p, q, u, v, w, y being VECTOR of V st p <> q & p,q // u,v & p,q // w,y holds u,v // w,y
t12_analoaf:: for V being RealLinearSpace for u, v, w, y being VECTOR of V st u,v // w,y holds ( v,u // y,w & w,y // u,v )
t13_analoaf:: for V being RealLinearSpace for u, v, w being VECTOR of V st u,v // v,w holds u,v // u,w
t14_analoaf:: for V being RealLinearSpace for u, v, w being VECTOR of V holds ( not u,v // u,w or u,v // v,w or u,w // w,v )
t15_analoaf:: for V being RealLinearSpace for v, u, y, w being VECTOR of V st v - u = y - w holds u,v // w,y
t16_analoaf:: for V being RealLinearSpace for y, v, w, u being VECTOR of V st y = (v + w) - u holds ( u,v // w,y & u,w // v,y )
t17_analoaf:: for V being RealLinearSpace st ex p, q being VECTOR of V st p <> q holds for u, v, w being VECTOR of V ex y being VECTOR of V st ( u,v // w,y & u,w // v,y & v <> y )
t18_analoaf:: for V being RealLinearSpace for p, v, w, u being VECTOR of V st p <> v & v,p // p,w holds ex y being VECTOR of V st ( u,p // p,y & u,v // w,y )
t19_analoaf:: for V being RealLinearSpace for u, v being VECTOR of V st ( for a, b being Real st (a * u) + (b * v) = 0. V holds ( a = 0 & b = 0 ) ) holds ( u <> v & u <> 0. V & v <> 0. V )
t2_analoaf:: for V being RealLinearSpace for y, u, v, w being VECTOR of V st y + u = v + w holds y - w = v - u
t20_analoaf:: for V being RealLinearSpace st ex u, v being VECTOR of V st for a, b being Real st (a * u) + (b * v) = 0. V holds ( a = 0 & b = 0 ) holds ex u, v, w, y being VECTOR of V st ( not u,v // w,y & not u,v // y,w )
t21_analoaf:: for V being RealLinearSpace st ex p, q being VECTOR of V st for w being VECTOR of V ex a, b being Real st (a * p) + (b * q) = w holds for u, v, w, y being VECTOR of V st not u,v // w,y & not u,v // y,w holds ex z being VECTOR of V st ( ( u,v // u,z or u,v // z,u ) & ( w,y // w,z or w,y // z,w ) )
t22_analoaf:: for V being RealLinearSpace for u, v, w, y being VECTOR of V holds ( [[u,v],[w,y]] in DirPar V iff u,v // w,y )
t23_analoaf:: for V being RealLinearSpace st ex u, v being VECTOR of V st for a, b being Real st (a * u) + (b * v) = 0. V holds ( a = 0 & b = 0 ) holds ( ex a, b being Element of (OASpace V) st a <> b & ( for a, b, c, d, p, q, r, s being Element of (OASpace V) holds ( a,b // c,c & ( a,b // b,a implies a = b ) & ( a <> b & a,b // p,q & a,b // r,s implies p,q // r,s ) & ( a,b // c,d implies b,a // d,c ) & ( a,b // b,c implies a,b // a,c ) & ( not a,b // a,c or a,b // b,c or a,c // c,b ) ) ) & ex a, b, c, d being Element of (OASpace V) st ( not a,b // c,d & not a,b // d,c ) & ( for a, b, c being Element of (OASpace V) ex d being Element of (OASpace V) st ( a,b // c,d & a,c // b,d & b <> d ) ) & ( for p, a, b, c being Element of (OASpace V) st p <> b & b,p // p,c holds ex d being Element of (OASpace V) st ( a,p // p,d & a,b // c,d ) ) )
t24_analoaf:: for V being RealLinearSpace st ex p, q being VECTOR of V st for w being VECTOR of V ex a, b being Real st (a * p) + (b * q) = w holds for a, b, c, d being Element of (OASpace V) st not a,b // c,d & not a,b // d,c holds ex t being Element of (OASpace V) st ( ( a,b // a,t or a,b // t,a ) & ( c,d // c,t or c,d // t,c ) )
t25_analoaf:: for AS being non empty AffinStruct holds ( ( ex a, b being Element of AS st a <> b & ( for a, b, c, d, p, q, r, s being Element of AS holds ( a,b // c,c & ( a,b // b,a implies a = b ) & ( a <> b & a,b // p,q & a,b // r,s implies p,q // r,s ) & ( a,b // c,d implies b,a // d,c ) & ( a,b // b,c implies a,b // a,c ) & ( not a,b // a,c or a,b // b,c or a,c // c,b ) ) ) & ex a, b, c, d being Element of AS st ( not a,b // c,d & not a,b // d,c ) & ( for a, b, c being Element of AS ex d being Element of AS st ( a,b // c,d & a,c // b,d & b <> d ) ) & ( for p, a, b, c being Element of AS st p <> b & b,p // p,c holds ex d being Element of AS st ( a,p // p,d & a,b // c,d ) ) ) iff AS is OAffinSpace ) by Def5, STRUCT_0:def_10;
t26_analoaf:: for V being RealLinearSpace st ex u, v being VECTOR of V st for a, b being Real st (a * u) + (b * v) = 0. V holds ( a = 0 & b = 0 ) holds OASpace V is OAffinSpace
t27_analoaf:: for AS being non empty AffinStruct holds ( ( ex a, b being Element of AS st a <> b & ( for a, b, c, d, p, q, r, s being Element of AS holds ( a,b // c,c & ( a,b // b,a implies a = b ) & ( a <> b & a,b // p,q & a,b // r,s implies p,q // r,s ) & ( a,b // c,d implies b,a // d,c ) & ( a,b // b,c implies a,b // a,c ) & ( not a,b // a,c or a,b // b,c or a,c // c,b ) ) ) & ex a, b, c, d being Element of AS st ( not a,b // c,d & not a,b // d,c ) & ( for a, b, c being Element of AS ex d being Element of AS st ( a,b // c,d & a,c // b,d & b <> d ) ) & ( for p, a, b, c being Element of AS st p <> b & b,p // p,c holds ex d being Element of AS st ( a,p // p,d & a,b // c,d ) ) & ( for a, b, c, d being Element of AS st not a,b // c,d & not a,b // d,c holds ex p being Element of AS st ( ( a,b // a,p or a,b // p,a ) & ( c,d // c,p or c,d // p,c ) ) ) ) iff AS is OAffinPlane ) by Def5, Def6, STRUCT_0:def_10;
t28_analoaf:: for V being RealLinearSpace st ex u, v being VECTOR of V st ( ( for a, b being Real st (a * u) + (b * v) = 0. V holds ( a = 0 & b = 0 ) ) & ( for w being VECTOR of V ex a, b being Real st w = (a * u) + (b * v) ) ) holds OASpace V is OAffinPlane
t3_analoaf:: for V being RealLinearSpace for u, v being VECTOR of V for a being Real holds a * (u - v) = - (a * (v - u))
t4_analoaf:: for V being RealLinearSpace for u, v being VECTOR of V for a, b being Real holds (a - b) * (u - v) = (b - a) * (v - u)
t5_analoaf:: for V being RealLinearSpace for u, v being VECTOR of V for a being Real st a <> 0 & a * u = v holds u = (a ") * v
t6_analoaf:: for V being RealLinearSpace for u, v being VECTOR of V for a being Real holds ( ( a <> 0 & a * u = v implies u = (a ") * v ) & ( a <> 0 & u = (a ") * v implies a * u = v ) )
t7_analoaf:: for V being RealLinearSpace for u, v, w, y being VECTOR of V st u,v // w,y & u <> v & w <> y holds ex a, b being Real st ( a * (v - u) = b * (y - w) & 0 < a & 0 < b )
t8_analoaf:: for V being RealLinearSpace for u, v being VECTOR of V holds u,v // u,v
t9_analoaf:: for V being RealLinearSpace for u, v, w being VECTOR of V holds ( u,v // w,w & u,u // v,w ) by Def1;
d1_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V holds Ortm (x,y,u) = ((pr1 (x,y,u)) * x) + ((- (pr2 (x,y,u))) * y);
d2_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V holds Orte (x,y,u) = ((pr2 (x,y,u)) * x) + ((- (pr1 (x,y,u))) * y);
d3_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V holds ( u,v,u1,v1 are_COrte_wrt x,y iff Orte (x,y,u), Orte (x,y,v) // u1,v1 );
d4_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V holds ( u,v,u1,v1 are_COrtm_wrt x,y iff Ortm (x,y,u), Ortm (x,y,v) // u1,v1 );
d5_analort:: for V being RealLinearSpace for x, y being VECTOR of V for b4 being Relation of [: the carrier of V, the carrier of V:] holds ( b4 = CORTE (V,x,y) iff for uu, vv being set holds ( [uu,vv] in b4 iff ex u1, u2, v1, v2 being VECTOR of V st ( uu = [u1,u2] & vv = [v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y ) ) );
d6_analort:: for V being RealLinearSpace for x, y being VECTOR of V for b4 being Relation of [: the carrier of V, the carrier of V:] holds ( b4 = CORTM (V,x,y) iff for uu, vv being set holds ( [uu,vv] in b4 iff ex u1, u2, v1, v2 being VECTOR of V st ( uu = [u1,u2] & vv = [v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y ) ) );
d7_analort:: for V being RealLinearSpace for x, y being VECTOR of V holds CESpace (V,x,y) = AffinStruct(# the carrier of V,(CORTE (V,x,y)) #);
d8_analort:: for V being RealLinearSpace for x, y being VECTOR of V holds CMSpace (V,x,y) = AffinStruct(# the carrier of V,(CORTM (V,x,y)) #);
t1_analort:: for V being RealLinearSpace for x, y, u, v being VECTOR of V st Gen x,y holds Ortm (x,y,(u + v)) = (Ortm (x,y,u)) + (Ortm (x,y,v))
t10_analort:: for V being RealLinearSpace for x, y, u, v being VECTOR of V st Gen x,y holds Orte (x,y,(u + v)) = (Orte (x,y,u)) + (Orte (x,y,v))
t11_analort:: for V being RealLinearSpace for x, y, u, v being VECTOR of V st Gen x,y holds Orte (x,y,(u - v)) = (Orte (x,y,u)) - (Orte (x,y,v))
t12_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V for n being Real st Gen x,y holds Orte (x,y,(n * u)) = n * (Orte (x,y,u))
t13_analort:: for V being RealLinearSpace for x, y, u, v being VECTOR of V st Gen x,y & Orte (x,y,u) = Orte (x,y,v) holds u = v
t14_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V st Gen x,y holds Orte (x,y,(Orte (x,y,u))) = - u
t15_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V st Gen x,y holds ex v being VECTOR of V st Orte (x,y,v) = u
t16_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y & u,v // u1,v1 holds Orte (x,y,u), Orte (x,y,v) // Orte (x,y,u1), Orte (x,y,v1)
t17_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y & u,v // u1,v1 holds Ortm (x,y,u), Ortm (x,y,v) // Ortm (x,y,u1), Ortm (x,y,v1)
t18_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrte_wrt x,y holds v,v1,u1,u are_COrte_wrt x,y
t19_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y holds v,v1,u,u1 are_COrtm_wrt x,y
t2_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V for n being Real st Gen x,y holds Ortm (x,y,(n * u)) = n * (Ortm (x,y,u))
t20_analort:: for V being RealLinearSpace for u, v, w, x, y being VECTOR of V holds u,u,v,w are_COrte_wrt x,y
t21_analort:: for V being RealLinearSpace for u, v, w, x, y being VECTOR of V holds u,u,v,w are_COrtm_wrt x,y
t22_analort:: for V being RealLinearSpace for u, v, w, x, y being VECTOR of V holds u,v,w,w are_COrte_wrt x,y
t23_analort:: for V being RealLinearSpace for u, v, w, x, y being VECTOR of V holds u,v,w,w are_COrtm_wrt x,y
t24_analort:: for V being RealLinearSpace for x, y, u, v being VECTOR of V st Gen x,y holds u,v, Orte (x,y,u), Orte (x,y,v) are_Ort_wrt x,y
t25_analort:: for V being RealLinearSpace for u, v, x, y being VECTOR of V holds u,v, Orte (x,y,u), Orte (x,y,v) are_COrte_wrt x,y
t26_analort:: for V being RealLinearSpace for u, v, x, y being VECTOR of V holds u,v, Ortm (x,y,u), Ortm (x,y,v) are_COrtm_wrt x,y
t27_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y holds ( u,v // u1,v1 iff ex u2, v2 being VECTOR of V st ( u2 <> v2 & u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y ) )
t28_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y holds ( u,v // u1,v1 iff ex u2, v2 being VECTOR of V st ( u2 <> v2 & u2,v2,u,v are_COrtm_wrt x,y & u2,v2,u1,v1 are_COrtm_wrt x,y ) )
t29_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y holds ( u,v,u1,v1 are_Ort_wrt x,y iff ( u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y ) )
t3_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds Ortm (x,y,(0. V)) = 0. V
t30_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrte_wrt x,y & u,v,v1,u1 are_COrte_wrt x,y & not u = v holds u1 = v1
t31_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1 being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,u1 are_COrtm_wrt x,y & not u = v holds u1 = v1
t32_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1, w being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrte_wrt x,y & u,v,u1,w are_COrte_wrt x,y & not u,v,v1,w are_COrte_wrt x,y holds u,v,w,v1 are_COrte_wrt x,y
t33_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1, w being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,u1,w are_COrtm_wrt x,y & not u,v,v1,w are_COrtm_wrt x,y holds u,v,w,v1 are_COrtm_wrt x,y
t34_analort:: for V being RealLinearSpace for u, v, u1, v1, x, y being VECTOR of V st u,v,u1,v1 are_COrte_wrt x,y holds v,u,v1,u1 are_COrte_wrt x,y
t35_analort:: for V being RealLinearSpace for u, v, u1, v1, x, y being VECTOR of V st u,v,u1,v1 are_COrtm_wrt x,y holds v,u,v1,u1 are_COrtm_wrt x,y
t36_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1, w being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrte_wrt x,y & u,v,v1,w are_COrte_wrt x,y holds u,v,u1,w are_COrte_wrt x,y
t37_analort:: for V being RealLinearSpace for x, y, u, v, u1, v1, w being VECTOR of V st Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,w are_COrtm_wrt x,y holds u,v,u1,w are_COrtm_wrt x,y
t38_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds for u, v, w being VECTOR of V ex u1 being VECTOR of V st ( w <> u1 & w,u1,u,v are_COrte_wrt x,y )
t39_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds for u, v, w being VECTOR of V ex u1 being VECTOR of V st ( w <> u1 & w,u1,u,v are_COrtm_wrt x,y )
t4_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V st Gen x,y holds Ortm (x,y,(- u)) = - (Ortm (x,y,u))
t40_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds for u, v, w being VECTOR of V ex u1 being VECTOR of V st ( w <> u1 & u,v,w,u1 are_COrte_wrt x,y )
t41_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds for u, v, w being VECTOR of V ex u1 being VECTOR of V st ( w <> u1 & u,v,w,u1 are_COrtm_wrt x,y )
t42_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1, w, w1, u2, v2 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrte_wrt x,y & w,w1,v,v1 are_COrte_wrt x,y & w,w1,u2,v2 are_COrte_wrt x,y & not w = w1 & not v = v1 holds u,u1,u2,v2 are_COrte_wrt x,y
t43_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1, w, w1, u2, v2 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & w,w1,v,v1 are_COrtm_wrt x,y & w,w1,u2,v2 are_COrtm_wrt x,y & not w = w1 & not v = v1 holds u,u1,u2,v2 are_COrtm_wrt x,y
t44_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1, w, w1, u2, v2 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrte_wrt x,y & v,v1,w,w1 are_COrte_wrt x,y & u2,v2,w,w1 are_COrte_wrt x,y & not u,u1,u2,v2 are_COrte_wrt x,y & not v = v1 holds w = w1
t45_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1, w, w1, u2, v2 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u2,v2,w,w1 are_COrtm_wrt x,y & not u,u1,u2,v2 are_COrtm_wrt x,y & not v = v1 holds w = w1
t46_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1, w, w1, u2, v2 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrte_wrt x,y & v,v1,w,w1 are_COrte_wrt x,y & u,u1,u2,v2 are_COrte_wrt x,y & not u2,v2,w,w1 are_COrte_wrt x,y & not v = v1 holds u = u1
t47_analort:: for V being RealLinearSpace for x, y, u, u1, v, v1, w, w1, u2, v2 being VECTOR of V st Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u,u1,u2,v2 are_COrtm_wrt x,y & not u2,v2,w,w1 are_COrtm_wrt x,y & not v = v1 holds u = u1
t48_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds for v, w, u1, v1, w1 being VECTOR of V st not v,v1,w,u1 are_COrte_wrt x,y & not v,v1,u1,w are_COrte_wrt x,y & u1,w1,u1,w are_COrte_wrt x,y holds ex u2 being VECTOR of V st ( ( v,v1,v,u2 are_COrte_wrt x,y or v,v1,u2,v are_COrte_wrt x,y ) & ( u1,w1,u1,u2 are_COrte_wrt x,y or u1,w1,u2,u1 are_COrte_wrt x,y ) )
t49_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds ex u, v, w being VECTOR of V st ( u,v,u,w are_COrte_wrt x,y & ( for v1, w1 being VECTOR of V holds ( not v1,w1,u,v are_COrte_wrt x,y or ( not v1,w1,u,w are_COrte_wrt x,y & not v1,w1,w,u are_COrte_wrt x,y ) or v1 = w1 ) ) )
t5_analort:: for V being RealLinearSpace for x, y, u, v being VECTOR of V st Gen x,y holds Ortm (x,y,(u - v)) = (Ortm (x,y,u)) - (Ortm (x,y,v))
t50_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds for v, w, u1, v1, w1 being VECTOR of V st not v,v1,w,u1 are_COrtm_wrt x,y & not v,v1,u1,w are_COrtm_wrt x,y & u1,w1,u1,w are_COrtm_wrt x,y holds ex u2 being VECTOR of V st ( ( v,v1,v,u2 are_COrtm_wrt x,y or v,v1,u2,v are_COrtm_wrt x,y ) & ( u1,w1,u1,u2 are_COrtm_wrt x,y or u1,w1,u2,u1 are_COrtm_wrt x,y ) )
t51_analort:: for V being RealLinearSpace for x, y being VECTOR of V st Gen x,y holds ex u, v, w being VECTOR of V st ( u,v,u,w are_COrtm_wrt x,y & ( for v1, w1 being VECTOR of V holds ( not v1,w1,u,v are_COrtm_wrt x,y or ( not v1,w1,u,w are_COrtm_wrt x,y & not v1,w1,w,u are_COrtm_wrt x,y ) or v1 = w1 ) ) )
t52_analort:: for V being RealLinearSpace for x, y being VECTOR of V for uu being set holds ( uu is Element of (CESpace (V,x,y)) iff uu is VECTOR of V ) ;
t53_analort:: for V being RealLinearSpace for x, y being VECTOR of V for uu being set holds ( uu is Element of (CMSpace (V,x,y)) iff uu is VECTOR of V ) ;
t54_analort:: for V being RealLinearSpace for u, v, u1, v1, x, y being VECTOR of V for p, q, r, s being Element of (CESpace (V,x,y)) st u = p & v = q & u1 = r & v1 = s holds ( p,q // r,s iff u,v,u1,v1 are_COrte_wrt x,y )
t55_analort:: for V being RealLinearSpace for u, v, u1, v1, x, y being VECTOR of V for p, q, r, s being Element of (CMSpace (V,x,y)) st u = p & v = q & u1 = r & v1 = s holds ( p,q // r,s iff u,v,u1,v1 are_COrtm_wrt x,y )
t6_analort:: for V being RealLinearSpace for x, y, u, v being VECTOR of V st Gen x,y & Ortm (x,y,u) = Ortm (x,y,v) holds u = v
t7_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V st Gen x,y holds Ortm (x,y,(Ortm (x,y,u))) = u
t8_analort:: for V being RealLinearSpace for x, y, u being VECTOR of V st Gen x,y holds ex v being VECTOR of V st u = Ortm (x,y,v)
t9_analort:: for V being RealLinearSpace for x, y, v being VECTOR of V st Gen x,y holds Orte (x,y,(- v)) = - (Orte (x,y,v))
d1_anproj_1:: for V being RealLinearSpace for p, q being Element of V holds ( are_Prop p,q iff ex a, b being Real st ( a * p = b * q & a <> 0 & b <> 0 ) );
d2_anproj_1:: for V being RealLinearSpace for u, v, w being Element of V holds ( u,v,w are_LinDep iff ex a, b, c being Real st ( ((a * u) + (b * v)) + (c * w) = 0. V & ( a <> 0 or b <> 0 or c <> 0 ) ) );
d3_anproj_1:: for V being RealLinearSpace for b2 being Equivalence_Relation of (NonZero V) holds ( b2 = Proportionality_as_EqRel_of V iff for x, y being set holds ( [x,y] in b2 iff ( x in NonZero V & y in NonZero V & ex u, v being Element of V st ( x = u & y = v & are_Prop u,v ) ) ) );
d4_anproj_1:: for V being RealLinearSpace for v being Element of V holds Dir v = Class ((Proportionality_as_EqRel_of V),v);
d5_anproj_1:: for V being RealLinearSpace for b2 being set holds ( b2 = ProjectivePoints V iff ex Y being Subset-Family of (NonZero V) st ( Y = Class (Proportionality_as_EqRel_of V) & b2 = Y ) );
d6_anproj_1:: for V being non trivial RealLinearSpace for b2 being Relation3 of ProjectivePoints V holds ( b2 = ProjectiveCollinearity V iff for x, y, z being set holds ( [x,y,z] in b2 iff ex p, q, r being Element of V st ( x = Dir p & y = Dir q & z = Dir r & not p is zero & not q is zero & not r is zero & p,q,r are_LinDep ) ) );
d7_anproj_1:: for V being non trivial RealLinearSpace holds ProjectiveSpace V = CollStr(# (ProjectivePoints V),(ProjectiveCollinearity V) #);
t1_anproj_1:: for V being RealLinearSpace for p, q being Element of V holds ( are_Prop p,q iff ex a being Real st ( a <> 0 & p = a * q ) )
t10_anproj_1:: for V being RealLinearSpace for u, v, w being Element of V st ( u = 0. V or v = 0. V or w = 0. V ) holds u,v,w are_LinDep
t11_anproj_1:: for V being RealLinearSpace for u, v, w being Element of V st ( are_Prop u,v or are_Prop w,u or are_Prop v,w ) holds w,u,v are_LinDep
t12_anproj_1:: for V being RealLinearSpace for u, v, w being Element of V st not u,v,w are_LinDep holds ( not u is zero & not v is zero & not w is zero & not are_Prop u,v & not are_Prop v,w & not are_Prop w,u )
t13_anproj_1:: for V being RealLinearSpace for p, q being Element of V st p + q = 0. V holds are_Prop p,q
t14_anproj_1:: for V being RealLinearSpace for p, q, u, v, w being Element of V st not are_Prop p,q & p,q,u are_LinDep & p,q,v are_LinDep & p,q,w are_LinDep & not p is zero & not q is zero holds u,v,w are_LinDep
t15_anproj_1:: for V being RealLinearSpace for u, v, w, p, q being Element of V st not u,v,w are_LinDep & u,v,p are_LinDep & v,w,q are_LinDep holds ex y being Element of V st ( u,w,y are_LinDep & p,q,y are_LinDep & not y is zero )
t16_anproj_1:: for V being RealLinearSpace for p, q being Element of V st not are_Prop p,q & not p is zero & not q is zero holds for u, v being Element of V ex y being Element of V st ( not y is zero & u,v,y are_LinDep & not are_Prop u,y & not are_Prop v,y )
t17_anproj_1:: for V being RealLinearSpace for p, q, r being Element of V st not p,q,r are_LinDep holds for u, v being Element of V st not u is zero & not v is zero & not are_Prop u,v holds ex y being Element of V st ( not y is zero & not u,v,y are_LinDep )
t18_anproj_1:: for V being RealLinearSpace for u, v, q, w, y, p, r being Element of V st u,v,q are_LinDep & w,y,q are_LinDep & u,w,p are_LinDep & v,y,p are_LinDep & u,y,r are_LinDep & v,w,r are_LinDep & p,q,r are_LinDep & not p is zero & not q is zero & not r is zero & not u,v,y are_LinDep & not u,v,w are_LinDep & not u,w,y are_LinDep holds v,w,y are_LinDep
t19_anproj_1:: for V being RealLinearSpace for x, y being set st [x,y] in Proportionality_as_EqRel_of V holds ( x is Element of V & y is Element of V )
t2_anproj_1:: for V being RealLinearSpace for p, u, q being Element of V st are_Prop p,u & are_Prop u,q holds are_Prop p,q
t20_anproj_1:: for V being RealLinearSpace for u, v being Element of V holds ( [u,v] in Proportionality_as_EqRel_of V iff ( not u is zero & not v is zero & are_Prop u,v ) )
t21_anproj_1:: for V being non trivial RealLinearSpace for p being Element of V st not p is zero holds Dir p is Element of ProjectivePoints V
t22_anproj_1:: for V being non trivial RealLinearSpace for p, q being Element of V st not p is zero & not q is zero holds ( Dir p = Dir q iff are_Prop p,q )
t23_anproj_1:: for V being non trivial RealLinearSpace holds ( the carrier of (ProjectiveSpace V) = ProjectivePoints V & the Collinearity of (ProjectiveSpace V) = ProjectiveCollinearity V ) ;
t24_anproj_1:: for x, y, z being set for V being non trivial RealLinearSpace st [x,y,z] in the Collinearity of (ProjectiveSpace V) holds ex p, q, r being Element of V st ( x = Dir p & y = Dir q & z = Dir r & not p is zero & not q is zero & not r is zero & p,q,r are_LinDep ) by Def6;
t25_anproj_1:: for V being non trivial RealLinearSpace for u, v, w being Element of V st not u is zero & not v is zero & not w is zero holds ( [(Dir u),(Dir v),(Dir w)] in the Collinearity of (ProjectiveSpace V) iff u,v,w are_LinDep )
t26_anproj_1:: for x being set for V being non trivial RealLinearSpace holds ( x is Element of (ProjectiveSpace V) iff ex u being Element of V st ( not u is zero & x = Dir u ) )
t3_anproj_1:: for V being RealLinearSpace for p being Element of V holds ( are_Prop p, 0. V iff p = 0. V )
t4_anproj_1:: for V being RealLinearSpace for u, u1, v, v1, w, w1 being Element of V st are_Prop u,u1 & are_Prop v,v1 & are_Prop w,w1 & u,v,w are_LinDep holds u1,v1,w1 are_LinDep
t5_anproj_1:: for V being RealLinearSpace for u, v, w being Element of V st u,v,w are_LinDep holds ( u,w,v are_LinDep & v,u,w are_LinDep & w,v,u are_LinDep & w,u,v are_LinDep & v,w,u are_LinDep )
t6_anproj_1:: for V being RealLinearSpace for v, w, u being Element of V st not v is zero & not w is zero & not are_Prop v,w holds ( v,w,u are_LinDep iff ex a, b being Real st u = (a * v) + (b * w) )
t7_anproj_1:: for V being RealLinearSpace for p, q being Element of V for a1, b1, a2, b2 being Real st not are_Prop p,q & (a1 * p) + (b1 * q) = (a2 * p) + (b2 * q) & not p is zero & not q is zero holds ( a1 = a2 & b1 = b2 )
t8_anproj_1:: for V being RealLinearSpace for u, v, w being Element of V for a1, b1, c1, a2, b2, c2 being Real st not u,v,w are_LinDep & ((a1 * u) + (b1 * v)) + (c1 * w) = ((a2 * u) + (b2 * v)) + (c2 * w) holds ( a1 = a2 & b1 = b2 & c1 = c2 )
t9_anproj_1:: for V being RealLinearSpace for p, q, u, v being Element of V for a1, b1, a2, b2 being Real st not are_Prop p,q & u = (a1 * p) + (b1 * q) & v = (a2 * p) + (b2 * q) & (a1 * b2) - (a2 * b1) = 0 & not p is zero & not q is zero & not are_Prop u,v & not u = 0. V holds v = 0. V
d1_anproj_2:: for V being RealLinearSpace for u, v, w being Element of V holds ( u,v,w are_Prop_Vect iff ( not u is zero & not v is zero & not w is zero ) );
d10_anproj_2:: for IT being non empty CollStr holds ( IT is at_least_3rank iff for p, q being Element of IT ex r being Element of IT st ( p <> r & q <> r & p,q,r is_collinear ) );
d11_anproj_2:: for IT being CollProjectiveSpace holds ( IT is Fanoian iff for p1, r2, q, r1, q1, p, r being Element of IT st p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear holds r2,r1,q1 is_collinear );
d12_anproj_2:: for IT being CollProjectiveSpace holds ( IT is Desarguesian iff for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being Element of IT st o <> q1 & p1 <> q1 & o <> q2 & p2 <> q2 & o <> q3 & p3 <> q3 & not o,p1,p2 is_collinear & not o,p1,p3 is_collinear & not o,p2,p3 is_collinear & p1,p2,r3 is_collinear & q1,q2,r3 is_collinear & p2,p3,r1 is_collinear & q2,q3,r1 is_collinear & p1,p3,r2 is_collinear & q1,q3,r2 is_collinear & o,p1,q1 is_collinear & o,p2,q2 is_collinear & o,p3,q3 is_collinear holds r1,r2,r3 is_collinear );
d13_anproj_2:: for IT being CollProjectiveSpace holds ( IT is Pappian iff for o, p1, p2, p3, q1, q2, q3, r1, r2, r3 being Element of IT st o <> p2 & o <> p3 & p2 <> p3 & p1 <> p2 & p1 <> p3 & o <> q2 & o <> q3 & q2 <> q3 & q1 <> q2 & q1 <> q3 & not o,p1,q1 is_collinear & o,p1,p2 is_collinear & o,p1,p3 is_collinear & o,q1,q2 is_collinear & o,q1,q3 is_collinear & p1,q2,r3 is_collinear & q1,p2,r3 is_collinear & p1,q3,r2 is_collinear & p3,q1,r2 is_collinear & p2,q3,r1 is_collinear & p3,q2,r1 is_collinear holds r1,r2,r3 is_collinear );
d14_anproj_2:: for IT being CollProjectiveSpace holds ( IT is 2-dimensional iff for p, p1, q, q1 being Element of IT ex r being Element of IT st ( p,p1,r is_collinear & q,q1,r is_collinear ) );
d15_anproj_2:: for IT being CollProjectiveSpace holds ( IT is at_most-3-dimensional iff for p, p1, q, q1, r2 being Element of IT ex r, r1 being Element of IT st ( p,q,r is_collinear & p1,q1,r1 is_collinear & r2,r,r1 is_collinear ) );
d2_anproj_2:: for V being RealLinearSpace for u, v, w, u1, v1, w1 being Element of V holds ( u,v,w,u1,v1,w1 lie_on_a_triangle iff ( u,v,w1 are_LinDep & u,w,v1 are_LinDep & v,w,u1 are_LinDep ) );
d3_anproj_2:: for V being RealLinearSpace for o, u, v, w, u2, v2, w2 being Element of V holds ( o,u,v,w,u2,v2,w2 are_perspective iff ( o,u,u2 are_LinDep & o,v,v2 are_LinDep & o,w,w2 are_LinDep ) );
d4_anproj_2:: for V being RealLinearSpace for o, u, v, w, u2, v2, w2 being Element of V holds ( o,u,v,w,u2,v2,w2 lie_on_an_angle iff ( not o,u,u2 are_LinDep & o,u,v are_LinDep & o,u,w are_LinDep & o,u2,v2 are_LinDep & o,u2,w2 are_LinDep ) );
d5_anproj_2:: for V being RealLinearSpace for o, u, v, w, u2, v2, w2 being Element of V holds ( o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop iff ( not are_Prop o,v & not are_Prop o,w & not are_Prop o,v2 & not are_Prop o,w2 & not are_Prop u,v & not are_Prop u,w & not are_Prop u2,v2 & not are_Prop u2,w2 & not are_Prop v,w & not are_Prop v2,w2 ) );
d6_anproj_2:: for IT being RealLinearSpace holds ( IT is up-3-dimensional iff ex u, v, w1 being Element of IT st for a, b, c being Real st ((a * u) + (b * v)) + (c * w1) = 0. IT holds ( a = 0 & b = 0 & c = 0 ) );
d7_anproj_2:: for CS being non empty CollStr holds ( CS is reflexive iff for p, q, r being Element of CS holds ( p,q,p is_collinear & p,p,q is_collinear & p,q,q is_collinear ) );
d8_anproj_2:: for CS being non empty CollStr holds ( CS is transitive iff for p, q, r, r1, r2 being Element of CS st p <> q & p,q,r is_collinear & p,q,r1 is_collinear & p,q,r2 is_collinear holds r,r1,r2 is_collinear );
d9_anproj_2:: for IT being non empty CollStr holds ( IT is Vebleian iff for p, p1, p2, r, r1 being Element of IT st p,p1,r is_collinear & p1,p2,r1 is_collinear holds ex r2 being Element of IT st ( p,p2,r2 is_collinear & r,r1,r2 is_collinear ) );
t1_anproj_2:: for V being RealLinearSpace for u, v, w being Element of V st ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds ( a = 0 & b = 0 & c = 0 ) ) holds ( not u is zero & not v is zero & not w is zero & not u,v,w are_LinDep & not are_Prop u,v )
t10_anproj_2:: for A being non empty set for x1 being Element of A ex f being Element of Funcs (A,REAL) st ( f . x1 = 1 & ( for z being set st z in A & z <> x1 holds f . z = 0 ) )
t11_anproj_2:: for A being non empty set for f, g, h being Element of Funcs (A,REAL) for x1, x2, x3 being Element of A st x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds h . z = 0 ) holds for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds ( a = 0 & b = 0 & c = 0 )
t12_anproj_2:: for A being non empty set for x1, x2, x3 being Element of A st x1 <> x2 & x1 <> x3 & x2 <> x3 holds ex f, g, h being Element of Funcs (A,REAL) st for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds ( a = 0 & b = 0 & c = 0 )
t13_anproj_2:: for A being non empty set for f, g, h being Element of Funcs (A,REAL) for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds h . z = 0 ) holds for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
t14_anproj_2:: for A being non empty set for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 holds ex f, g, h being Element of Funcs (A,REAL) st for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))
t15_anproj_2:: for A being non empty set for x1, x2, x3 being Element of A st A = {x1,x2,x3} & x1 <> x2 & x1 <> x3 & x2 <> x3 holds ex f, g, h being Element of Funcs (A,REAL) st ( ( for a, b, c being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) = RealFuncZero A holds ( a = 0 & b = 0 & c = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h])) ) )
t16_anproj_2:: ex V being non trivial RealLinearSpace ex u, v, w being Element of V st ( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds ( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) )
t17_anproj_2:: for A being non empty set for f, g, h, f1 being Element of Funcs (A,REAL) for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds f1 . z = 0 ) holds for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds ( a = 0 & b = 0 & c = 0 & d = 0 )
t18_anproj_2:: for A being non empty set for x1, x2, x3, x4 being Element of A st x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds ex f, g, h, f1 being Element of Funcs (A,REAL) st for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds ( a = 0 & b = 0 & c = 0 & d = 0 )
t19_anproj_2:: for A being non empty set for f, g, h, f1 being Element of Funcs (A,REAL) for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 & f . x1 = 1 & ( for z being set st z in A & z <> x1 holds f . z = 0 ) & g . x2 = 1 & ( for z being set st z in A & z <> x2 holds g . z = 0 ) & h . x3 = 1 & ( for z being set st z in A & z <> x3 holds h . z = 0 ) & f1 . x4 = 1 & ( for z being set st z in A & z <> x4 holds f1 . z = 0 ) holds for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
t2_anproj_2:: for V being RealLinearSpace for u, v, u1, v1 being Element of V st ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds ( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) holds ( not u is zero & not v is zero & not are_Prop u,v & not u1 is zero & not v1 is zero & not are_Prop u1,v1 & not u,v,u1 are_LinDep & not u1,v1,u are_LinDep )
t20_anproj_2:: for A being non empty set for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds ex f, g, h, f1 being Element of Funcs (A,REAL) st for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1]))
t21_anproj_2:: for A being non empty set for x1, x2, x3, x4 being Element of A st A = {x1,x2,x3,x4} & x1 <> x2 & x1 <> x3 & x1 <> x4 & x2 <> x3 & x2 <> x4 & x3 <> x4 holds ex f, g, h, f1 being Element of Funcs (A,REAL) st ( ( for a, b, c, d being Real st (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) = RealFuncZero A holds ( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for h9 being Element of Funcs (A,REAL) ex a, b, c, d being Real st h9 = (RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncAdd A) . (((RealFuncExtMult A) . [a,f]),((RealFuncExtMult A) . [b,g]))),((RealFuncExtMult A) . [c,h]))),((RealFuncExtMult A) . [d,f1])) ) )
t22_anproj_2:: ex V being non trivial RealLinearSpace ex u, v, w, u1 being Element of V st ( ( for a, b, c, d being Real st (((a * u) + (b * v)) + (c * w)) + (d * u1) = 0. V holds ( a = 0 & b = 0 & c = 0 & d = 0 ) ) & ( for y being Element of V ex a, b, c, d being Real st y = (((a * u) + (b * v)) + (c * w)) + (d * u1) ) )
t23_anproj_2:: for V being non trivial RealLinearSpace for p, q, r being Element of (ProjectiveSpace V) holds ( p,q,r is_collinear iff ex u, v, w being Element of V st ( p = Dir u & q = Dir v & r = Dir w & not u is zero & not v is zero & not w is zero & u,v,w are_LinDep ) )
t24_anproj_2:: for V being non trivial RealLinearSpace for p, q, r being Element of (ProjectiveSpace V) st p,q,r is_collinear holds ( p,r,q is_collinear & q,p,r is_collinear & r,q,p is_collinear & r,p,q is_collinear & q,r,p is_collinear )
t25_anproj_2:: for V being non trivial RealLinearSpace st ex u, v being Element of V st for a, b being Real st (a * u) + (b * v) = 0. V holds ( a = 0 & b = 0 ) holds ProjectiveSpace V is at_least_3rank
t26_anproj_2:: for V being non trivial RealLinearSpace for p1, r2, q, r1, q1, p, r being Element of (ProjectiveSpace V) st p1,r2,q is_collinear & r1,q1,q is_collinear & p1,r1,p is_collinear & r2,q1,p is_collinear & p1,q1,r is_collinear & r2,r1,r is_collinear & p,q,r is_collinear & not p1,r2,q1 is_collinear & not p1,r2,r1 is_collinear & not p1,r1,q1 is_collinear holds r2,r1,q1 is_collinear
t27_anproj_2:: for V being non trivial RealLinearSpace st ex u, v, w being Element of V st ( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds ( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) holds ex x1, x2 being Element of (ProjectiveSpace V) st ( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st ( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) )
t28_anproj_2:: for V being non trivial RealLinearSpace st ex x1, x2 being Element of (ProjectiveSpace V) st ( x1 <> x2 & ( for r1, r2 being Element of (ProjectiveSpace V) ex q being Element of (ProjectiveSpace V) st ( x1,x2,q is_collinear & r1,r2,q is_collinear ) ) ) holds for p, p1, q, q1 being Element of (ProjectiveSpace V) ex r being Element of (ProjectiveSpace V) st ( p,p1,r is_collinear & q,q1,r is_collinear )
t29_anproj_2:: for V being non trivial RealLinearSpace st ex u, v, w being Element of V st ( ( for a, b, c being Real st ((a * u) + (b * v)) + (c * w) = 0. V holds ( a = 0 & b = 0 & c = 0 ) ) & ( for y being Element of V ex a, b, c being Real st y = ((a * u) + (b * v)) + (c * w) ) ) holds ex CS being CollProjectiveSpace st ( CS = ProjectiveSpace V & CS is 2-dimensional )
t3_anproj_2:: for V being RealLinearSpace for p, q, r being Element of V st ( for w being Element of V ex a, b, c being Real st w = ((a * p) + (b * q)) + (c * r) ) & ( for a, b, c being Real st ((a * p) + (b * q)) + (c * r) = 0. V holds ( a = 0 & b = 0 & c = 0 ) ) holds for u, u1 being Element of V ex y being Element of V st ( p,q,y are_LinDep & u,u1,y are_LinDep & not y is zero )
t30_anproj_2:: for V being non trivial RealLinearSpace st ex y, u, v, w being Element of V st ( ( for w1 being Element of V ex a, b, a1, b1 being Real st w1 = (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds ( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) holds ex p, q1, q2 being Element of (ProjectiveSpace V) st ( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st ( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) )
t31_anproj_2:: for V being non trivial RealLinearSpace st ProjectiveSpace V is proper & ProjectiveSpace V is at_least_3rank & ex p, q1, q2 being Element of (ProjectiveSpace V) st ( not p,q1,q2 is_collinear & ( for r1, r2 being Element of (ProjectiveSpace V) ex q3, r3 being Element of (ProjectiveSpace V) st ( r1,r2,r3 is_collinear & q1,q2,q3 is_collinear & p,r3,q3 is_collinear ) ) ) holds ex CS being CollProjectiveSpace st ( CS = ProjectiveSpace V & CS is at_most-3-dimensional )
t32_anproj_2:: for V being non trivial RealLinearSpace st ex y, u, v, w being Element of V st ( ( for w1 being Element of V ex a, b, c, c1 being Real st w1 = (((a * y) + (b * u)) + (c * v)) + (c1 * w) ) & ( for a, b, a1, b1 being Real st (((a * y) + (b * u)) + (a1 * v)) + (b1 * w) = 0. V holds ( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) holds ex CS being CollProjectiveSpace st ( CS = ProjectiveSpace V & CS is at_most-3-dimensional )
t33_anproj_2:: for V being non trivial RealLinearSpace st ex u, v, u1, v1 being Element of V st for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds ( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) holds ex CS being CollProjectiveSpace st ( CS = ProjectiveSpace V & not CS is 2-dimensional )
t34_anproj_2:: for V being non trivial RealLinearSpace st ex u, v, u1, v1 being Element of V st ( ( for w being Element of V ex a, b, a1, b1 being Real st w = (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) ) & ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds ( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) ) holds ex CS being CollProjectiveSpace st ( CS = ProjectiveSpace V & CS is up-3-dimensional & CS is at_most-3-dimensional )
t35_anproj_2:: for CS being non empty CollStr holds ( CS is 2-dimensional CollProjectiveSpace iff ( CS is proper at_least_3rank CollSp & ( for p, p1, q, q1 being Element of CS ex r being Element of CS st ( p,p1,r is_collinear & q,q1,r is_collinear ) ) ) )
t4_anproj_2:: for V being RealLinearSpace for p, q, r, s being Element of V st ( for w being Element of V ex a, b, c, d being Real st w = (((a * p) + (b * q)) + (c * r)) + (d * s) ) & ( for a, b, c, d being Real st (((a * p) + (b * q)) + (c * r)) + (d * s) = 0. V holds ( a = 0 & b = 0 & c = 0 & d = 0 ) ) holds for u, v being Element of V st not u is zero & not v is zero holds ex y, w being Element of V st ( u,v,w are_LinDep & q,r,y are_LinDep & p,w,y are_LinDep & not y is zero & not w is zero )
t5_anproj_2:: for V being RealLinearSpace for u, v, u1, v1 being Element of V st ( for a, b, a1, b1 being Real st (((a * u) + (b * v)) + (a1 * u1)) + (b1 * v1) = 0. V holds ( a = 0 & b = 0 & a1 = 0 & b1 = 0 ) ) holds for y being Element of V holds ( y is zero or not u,v,y are_LinDep or not u1,v1,y are_LinDep )
t6_anproj_2:: for V being RealLinearSpace for o, u, u2 being Element of V st o,u,u2 are_LinDep & not are_Prop o,u & not are_Prop o,u2 & not are_Prop u,u2 & o,u,u2 are_Prop_Vect holds ( ex a1, b1 being Real st ( b1 * u2 = o + (a1 * u) & a1 <> 0 & b1 <> 0 ) & ex a2, c2 being Real st ( u2 = (c2 * o) + (a2 * u) & c2 <> 0 & a2 <> 0 ) )
t7_anproj_2:: for V being RealLinearSpace for p, q, r being Element of V st p,q,r are_LinDep & not are_Prop p,q & p,q,r are_Prop_Vect holds ex a, b being Real st r = (a * p) + (b * q)
t8_anproj_2:: for V being RealLinearSpace for o, u, v, w, u2, v2, w2, u1, v1, w1 being Element of V st not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 are_perspective & not are_Prop o,u2 & not are_Prop o,v2 & not are_Prop o,w2 & not are_Prop u,u2 & not are_Prop v,v2 & not are_Prop w,w2 & not o,u,v are_LinDep & not o,u,w are_LinDep & not o,v,w are_LinDep & u,v,w,u1,v1,w1 lie_on_a_triangle & u2,v2,w2,u1,v1,w1 lie_on_a_triangle holds u1,v1,w1 are_LinDep
t9_anproj_2:: for V being RealLinearSpace for o, u, v, w, u2, v2, w2, u1, v1, w1 being Element of V st not o is zero & u,v,w are_Prop_Vect & u2,v2,w2 are_Prop_Vect & u1,v1,w1 are_Prop_Vect & o,u,v,w,u2,v2,w2 lie_on_an_angle & o,u,v,w,u2,v2,w2 are_half_mutually_not_Prop & u,v2,w1 are_LinDep & u2,v,w1 are_LinDep & u,w2,v1 are_LinDep & w,u2,v1 are_LinDep & v,w2,u1 are_LinDep & w,v2,u1 are_LinDep holds u1,v1,w1 are_LinDep
d1_aofa_000:: for f being homogeneous Function for x being set holds ( x is_a_unity_wrt f iff for y, z being set st ( <*y,z*> in dom f or <*z,y*> in dom f ) holds ( <*x,y*> in dom f & f . <*x,y*> = y & <*y,x*> in dom f & f . <*y,x*> = y ) );
d10_aofa_000:: for S being non empty UAStr holds ( S is with_empty-instruction iff ( 1 in dom the charact of S & the charact of S . 1 is non empty homogeneous quasi_total 0 -ary PartFunc of ( the carrier of S *), the carrier of S ) );
d11_aofa_000:: for S being non empty UAStr holds ( S is with_catenation iff ( 2 in dom the charact of S & the charact of S . 2 is non empty homogeneous quasi_total 2 -ary PartFunc of ( the carrier of S *), the carrier of S ) );
d12_aofa_000:: for S being non empty UAStr holds ( S is with_if-instruction iff ( 3 in dom the charact of S & the charact of S . 3 is non empty homogeneous quasi_total 3 -ary PartFunc of ( the carrier of S *), the carrier of S ) );
d13_aofa_000:: for S being non empty UAStr holds ( S is with_while-instruction iff ( 4 in dom the charact of S & the charact of S . 4 is non empty homogeneous quasi_total 2 -ary PartFunc of ( the carrier of S *), the carrier of S ) );
d14_aofa_000:: for S being non empty UAStr holds ( S is associative iff the charact of S . 2 is non empty homogeneous quasi_total 2 -ary associative PartFunc of ( the carrier of S *), the carrier of S );
d15_aofa_000:: for S being non-empty UAStr holds ( S is unital iff ex f being non empty homogeneous quasi_total 2 -ary PartFunc of ( the carrier of S *), the carrier of S st ( f = the charact of S . 2 & (Den ((In (1,(dom the charact of S))),S)) . {} is_a_unity_wrt f ) );
d16_aofa_000:: for A being non-empty with_empty-instruction UAStr holds EmptyIns A = (Den ((In (1,(dom the charact of A))),A)) . {};
d17_aofa_000:: for A being non-empty with_catenation UAStr for I1, I2 being Algorithm of A holds I1 \; I2 = (Den ((In (2,(dom the charact of A))),A)) . <*I1,I2*>;
d18_aofa_000:: for A being non-empty with_if-instruction UAStr for C, I1, I2 being Algorithm of A holds if-then-else (C,I1,I2) = (Den ((In (3,(dom the charact of A))),A)) . <*C,I1,I2*>;
d19_aofa_000:: for A being non-empty with_empty-instruction with_if-instruction UAStr for C, I being Algorithm of A holds if-then (C,I) = if-then-else (C,I,(EmptyIns A));
d2_aofa_000:: for f being homogeneous Function holds ( f is associative iff for x, y, z being set st <*x,y*> in dom f & <*y,z*> in dom f & <*(f . <*x,y*>),z*> in dom f & <*x,(f . <*y,z*>)*> in dom f holds f . <*(f . <*x,y*>),z*> = f . <*x,(f . <*y,z*>)*> );
d20_aofa_000:: for A being non-empty with_while-instruction UAStr for C, I being Algorithm of A holds while (C,I) = (Den ((In (4,(dom the charact of A))),A)) . <*C,I*>;
d21_aofa_000:: for A being preIfWhileAlgebra for I0, C, I, J being Element of A holds for-do (I0,C,J,I) = I0 \; (while (C,(I \; J)));
d22_aofa_000:: for A being preIfWhileAlgebra holds ElementaryInstructions A = ((( the carrier of A \ {(EmptyIns A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (I1 \; I2) where I1, I2 is Algorithm of A : ( I1 <> I1 \; I2 & I2 <> I1 \; I2 ) } ;
d23_aofa_000:: for A being preIfWhileAlgebra holds ( A is infinite iff ElementaryInstructions A is infinite );
d24_aofa_000:: for A being preIfWhileAlgebra holds ( A is degenerated iff ( ex I1, I2 being Element of A st ( ( I1 <> EmptyIns A & I1 \; I2 = I2 ) or ( I2 <> EmptyIns A & I1 \; I2 = I1 ) or ( ( I1 <> EmptyIns A or I2 <> EmptyIns A ) & I1 \; I2 = EmptyIns A ) ) or ex C, I1, I2 being Element of A st if-then-else (C,I1,I2) = EmptyIns A or ex C, I being Element of A st while (C,I) = EmptyIns A or ex I1, I2, C, J1, J2 being Element of A st ( I1 <> EmptyIns A & I2 <> EmptyIns A & I1 \; I2 = if-then-else (C,J1,J2) ) or ex I1, I2, C, J being Element of A st ( I1 <> EmptyIns A & I2 <> EmptyIns A & I1 \; I2 = while (C,J) ) or ex C1, I1, I2, C2, J being Element of A st if-then-else (C1,I1,I2) = while (C2,J) ) );
d25_aofa_000:: for A being preIfWhileAlgebra holds ( A is well_founded iff ElementaryInstructions A is GeneratorSet of A );
d26_aofa_000:: ECIW-signature = <*0,2*> ^ <*3,2*>;
d27_aofa_000:: for A being non empty partial non-empty UAStr holds ( A is ECIW-strict iff signature A = ECIW-signature );
d28_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for f being Function of [:S, the carrier of A:],S holds ( f is complying_with_empty-instruction iff for s being Element of S holds f . (s,(EmptyIns A)) = s );
d29_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for f being Function of [:S, the carrier of A:],S holds ( f is complying_with_catenation iff for s being Element of S for I1, I2 being Element of A holds f . (s,(I1 \; I2)) = f . ((f . (s,I1)),I2) );
d3_aofa_000:: for f being homogeneous Function holds ( f is unital iff ex x being set st x is_a_unity_wrt f );
d30_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being Function of [:S, the carrier of A:],S holds ( f complies_with_if_wrt T iff for s being Element of S for C, I1, I2 being Element of A holds ( ( f . (s,C) in T implies f . (s,(if-then-else (C,I1,I2))) = f . ((f . (s,C)),I1) ) & ( f . (s,C) nin T implies f . (s,(if-then-else (C,I1,I2))) = f . ((f . (s,C)),I2) ) ) );
d31_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being Function of [:S, the carrier of A:],S holds ( f complies_with_while_wrt T iff for s being Element of S for C, I being Element of A holds ( ( f . (s,C) in T implies f . (s,(while (C,I))) = f . ((f . ((f . (s,C)),I)),(while (C,I))) ) & ( f . (s,C) nin T implies f . (s,(while (C,I))) = f . (s,C) ) ) );
d32_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for b4 being Function of [:S, the carrier of A:],S holds ( b4 is ExecutionFunction of A,S,T iff ( b4 is complying_with_empty-instruction & b4 is complying_with_catenation & b4 complies_with_if_wrt T & b4 complies_with_while_wrt T ) );
d33_aofa_000:: for A being preIfWhileAlgebra for I being Element of A for S being non empty set for s being Element of S for T being Subset of S for f being ExecutionFunction of A,S,T holds ( f iteration_terminates_for I,s iff ex r being non empty FinSequence of S st ( r . 1 = s & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds ( r . i in T & r . (i + 1) = f . ((r . i),I) ) ) ) );
d34_aofa_000:: for A being preIfWhileAlgebra for I being Element of A for S being non empty set for s being Element of S for T being Subset of S for f being ExecutionFunction of A,S,T for b7 being R_eal holds ( ( f iteration_terminates_for I,s implies ( b7 = iteration-degree (I,s,f) iff ex r being non empty FinSequence of S st ( b7 = (len r) - 1 & r . 1 = s & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds ( r . i in T & r . (i + 1) = f . ((r . i),I) ) ) ) ) ) & ( not f iteration_terminates_for I,s implies ( b7 = iteration-degree (I,s,f) iff b7 = +infty ) ) );
d35_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for b5 being Subset of [:S, the carrier of A:] holds ( b5 = TerminatingPrograms (A,S,T,f) iff ( [:S,(ElementaryInstructions A):] c= b5 & [:S,{(EmptyIns A)}:] c= b5 & ( for s being Element of S for C, I, J being Element of A holds ( ( [s,I] in b5 & [(f . (s,I)),J] in b5 implies [s,(I \; J)] in b5 ) & ( [s,C] in b5 & [(f . (s,C)),I] in b5 & f . (s,C) in T implies [s,(if-then-else (C,I,J))] in b5 ) & ( [s,C] in b5 & [(f . (s,C)),J] in b5 & f . (s,C) nin T implies [s,(if-then-else (C,I,J))] in b5 ) & ( [s,C] in b5 & ex r being non empty FinSequence of S st ( r . 1 = f . (s,C) & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds ( r . i in T & [(r . i),(I \; C)] in b5 & r . (i + 1) = f . ((r . i),(I \; C)) ) ) ) implies [s,(while (C,I))] in b5 ) ) ) & ( for P being Subset of [:S, the carrier of A:] st [:S,(ElementaryInstructions A):] c= P & [:S,{(EmptyIns A)}:] c= P & ( for s being Element of S for C, I, J being Element of A holds ( ( [s,I] in P & [(f . (s,I)),J] in P implies [s,(I \; J)] in P ) & ( [s,C] in P & [(f . (s,C)),I] in P & f . (s,C) in T implies [s,(if-then-else (C,I,J))] in P ) & ( [s,C] in P & [(f . (s,C)),J] in P & f . (s,C) nin T implies [s,(if-then-else (C,I,J))] in P ) & ( [s,C] in P & ex r being non empty FinSequence of S st ( r . 1 = f . (s,C) & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds ( r . i in T & [(r . i),(I \; C)] in P & r . (i + 1) = f . ((r . i),(I \; C)) ) ) ) implies [s,(while (C,I))] in P ) ) ) holds b5 c= P ) ) );
d36_aofa_000:: for A being preIfWhileAlgebra for I being Element of A holds ( I is absolutely-terminating iff for S being non empty set for s being Element of S for T being Subset of S for f being ExecutionFunction of A,S,T holds [s,I] in TerminatingPrograms (A,S,T,f) );
d37_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for I being Element of A for f being ExecutionFunction of A,S,T holds ( I is_terminating_wrt f iff for s being Element of S holds [s,I] in TerminatingPrograms (A,S,T,f) );
d38_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for I being Element of A for f being ExecutionFunction of A,S,T for Z being set holds ( I is_terminating_wrt f,Z iff for s being Element of S st s in Z holds [s,I] in TerminatingPrograms (A,S,T,f) );
d39_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for I being Element of A for f being ExecutionFunction of A,S,T for Z being set holds ( Z is_invariant_wrt I,f iff for s being Element of S st s in Z holds f . (s,I) in Z );
d4_aofa_000:: for f, g being Function for X being set holds (f,X) +* g = g +* (f | X);
d5_aofa_000:: for f being Function for x being set holds f orbit x = { ((iter (f,n)) . x) where n is Element of NAT : x in dom (iter (f,n)) } ;
d6_aofa_000:: for f being Function st rng f c= dom f holds for A, x being set for b4 being Function holds ( b4 = (A,x) iter f iff ( dom b4 = dom f & ( for a being set st a in dom f holds ( ( f orbit a c= A implies b4 . a = x ) & ( for n being Nat st (iter (f,n)) . a nin A & ( for i being Nat st i < n holds (iter (f,i)) . a in A ) holds b4 . a = (iter (f,n)) . a ) ) ) ) );
d7_aofa_000:: for f being Function st rng f c= dom f holds for A being set for g, b4 being Function holds ( b4 = (A,g) iter f iff ( dom b4 = dom f & ( for a being set st a in dom f holds ( ( f orbit a c= A implies b4 . a = g . a ) & ( for n being Nat st (iter (f,n)) . a nin A & ( for i being Nat st i < n holds (iter (f,i)) . a in A ) holds b4 . a = (iter (f,n)) . a ) ) ) ) );
d8_aofa_000:: for A being non-empty UAStr for B being Subset of A for n being Nat for b4 being Subset of A holds ( b4 = B |^ n iff ex F being Function of NAT,(bool the carrier of A) st ( b4 = F . n & F . 0 = B & ( for n being Nat holds F . (n + 1) = (F . n) \/ { ((Den (o,A)) . p) where o is Element of dom the charact of A, p is Element of the carrier of A * : ( p in dom (Den (o,A)) & rng p c= F . n ) } ) ) );
d9_aofa_000:: for A being Universal_Algebra holds Generators A = the carrier of A \ (union { (rng o) where o is Element of Operations A : verum } );
s1_aofa_000:: scheme MaxVal{ F1() -> non empty set , F2() -> set , P1[ set , set ] } : ex n being Nat st for x being Element of F1() st x in F2() holds P1[x,n] provided A1: F2() is finite and A2: for x being Element of F1() st x in F2() holds ex n being Nat st P1[x,n] and A3: for x being Element of F1() for n, m being Nat st P1[x,n] & n <= m holds P1[x,m]
s2_aofa_000:: scheme StructInd{ F1() -> well_founded ECIW-strict preIfWhileAlgebra, F2() -> Element of F1(), P1[ set ] } : P1[F2()] provided A1: for I being Element of F1() st I in ElementaryInstructions F1() holds P1[I] and A2: P1[ EmptyIns F1()] and A3: for I1, I2 being Element of F1() st P1[I1] & P1[I2] holds P1[I1 \; I2] and A4: for C, I1, I2 being Element of F1() st P1[C] & P1[I1] & P1[I2] holds P1[ if-then-else (C,I1,I2)] and A5: for C, I being Element of F1() st P1[C] & P1[I] holds P1[ while (C,I)]
s3_aofa_000:: scheme Termination{ F1() -> preIfWhileAlgebra, F2() -> Element of F1(), F3() -> non empty set , F4() -> Element of F3(), F5() -> Subset of F3(), F6() -> ExecutionFunction of F1(),F3(),F5(), F7( set ) -> Nat, P1[ set ] } : F6() iteration_terminates_for F2(),F4() provided A1: ( F4() in F5() iff P1[F4()] ) and A2: for s being Element of F3() st P1[s] holds ( ( P1[F6() . (s,F2())] implies F6() . (s,F2()) in F5() ) & ( F6() . (s,F2()) in F5() implies P1[F6() . (s,F2())] ) & F7((F6() . (s,F2()))) < F7(s) )
s4_aofa_000:: scheme Termination2{ F1() -> preIfWhileAlgebra, F2() -> Element of F1(), F3() -> non empty set , F4() -> Element of F3(), F5() -> Subset of F3(), F6() -> ExecutionFunction of F1(),F3(),F5(), F7( set ) -> Nat, P1[ set ], P2[ set ] } : F6() iteration_terminates_for F2(),F4() provided A1: P1[F4()] and A2: ( F4() in F5() iff P2[F4()] ) and A3: for s being Element of F3() st P1[s] & s in F5() & P2[s] holds ( P1[F6() . (s,F2())] & ( P2[F6() . (s,F2())] implies F6() . (s,F2()) in F5() ) & ( F6() . (s,F2()) in F5() implies P2[F6() . (s,F2())] ) & F7((F6() . (s,F2()))) < F7(s) )
s5_aofa_000:: scheme InvariantSch{ F1() -> preIfWhileAlgebra, F2() -> Element of F1(), F3() -> Element of F1(), F4() -> non empty set , F5() -> Element of F4(), F6() -> Subset of F4(), F7() -> ExecutionFunction of F1(),F4(),F6(), P1[ set ], P2[ set ] } : ( P1[F7() . (F5(),(while (F2(),F3())))] & P2[F7() . (F5(),(while (F2(),F3())))] ) provided A1: P1[F5()] and A2: F7() iteration_terminates_for F3() \; F2(),F7() . (F5(),F2()) and A3: for s being Element of F4() st P1[s] & s in F6() & P2[s] holds P1[F7() . (s,F3())] and A4: for s being Element of F4() st P1[s] holds ( P1[F7() . (s,F2())] & ( F7() . (s,F2()) in F6() implies P2[F7() . (s,F2())] ) & ( P2[F7() . (s,F2())] implies F7() . (s,F2()) in F6() ) )
s6_aofa_000:: scheme coInvariantSch{ F1() -> preIfWhileAlgebra, F2() -> Element of F1(), F3() -> Element of F1(), F4() -> non empty set , F5() -> Element of F4(), F6() -> Subset of F4(), F7() -> ExecutionFunction of F1(),F4(),F6(), P1[ set ] } : P1[F5()] provided A1: P1[F7() . (F5(),(while (F2(),F3())))] and A2: F7() iteration_terminates_for F3() \; F2(),F7() . (F5(),F2()) and A3: for s being Element of F4() st P1[F7() . ((F7() . (s,F2())),F3())] & F7() . (s,F2()) in F6() holds P1[F7() . (s,F2())] and A4: for s being Element of F4() st P1[F7() . (s,F2())] holds P1[s]
s7_aofa_000:: scheme IndDef{ F1() -> free ECIW-strict preIfWhileAlgebra, F2() -> non empty set , F3() -> Element of F2(), F4( set ) -> set , F5( set , set ) -> Element of F2(), F6( set , set ) -> Element of F2(), F7( set , set , set ) -> Element of F2() } : ex f being Function of the carrier of F1(),F2() st ( ( for I being Element of F1() st I in ElementaryInstructions F1() holds f . I = F4(I) ) & f . (EmptyIns F1()) = F3() & ( for I1, I2 being Element of F1() holds f . (I1 \; I2) = F5((f . I1),(f . I2)) ) & ( for C, I1, I2 being Element of F1() holds f . (if-then-else (C,I1,I2)) = F7((f . C),(f . I1),(f . I2)) ) & ( for C, I being Element of F1() holds f . (while (C,I)) = F6((f . C),(f . I)) ) ) provided A1: for I being Element of F1() st I in ElementaryInstructions F1() holds F4(I) in F2()
t1_aofa_000:: for f, g, h being Function for A being set st A c= dom f & A c= dom g & rng h c= A & ( for x being set st x in A holds f . x = g . x ) holds f * h = g * h
t10_aofa_000:: for f, g being Function for a, A being set st rng f c= dom f & a in dom f & not f orbit a c= A holds ex n being Nat st ( ((A,g) iter f) . a = (iter (f,n)) . a & (iter (f,n)) . a nin A & ( for i being Nat st i < n holds (iter (f,i)) . a in A ) )
t100_aofa_000:: for A being preIfWhileAlgebra for C, I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T st A is free & [s,(while (C,I))] in TerminatingPrograms (A,S,T,f) & f . (s,C) in T holds [(f . (s,C)),I] in TerminatingPrograms (A,S,T,f)
t101_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T for C, I being absolutely-terminating Element of A st f iteration_terminates_for I \; C,f . (s,C) holds [s,(while (C,I))] in TerminatingPrograms (A,S,T,f)
t102_aofa_000:: for S being non empty set for T being Subset of S for A being free ECIW-strict preIfWhileAlgebra for f1, f2 being ExecutionFunction of A,S,T st f1 | [:S,(ElementaryInstructions A):] = f2 | [:S,(ElementaryInstructions A):] holds TerminatingPrograms (A,S,T,f1) = TerminatingPrograms (A,S,T,f2)
t103_aofa_000:: for S being non empty set for T being Subset of S for A being free ECIW-strict preIfWhileAlgebra for f1, f2 being ExecutionFunction of A,S,T st f1 | [:S,(ElementaryInstructions A):] = f2 | [:S,(ElementaryInstructions A):] holds for s being Element of S for I being Element of A st [s,I] in TerminatingPrograms (A,S,T,f1) holds f1 . (s,I) = f2 . (s,I) by Lm3;
t104_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for I being absolutely-terminating Element of A holds I is_terminating_wrt f
t105_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for I being Element of A holds ( I is_terminating_wrt f iff I is_terminating_wrt f,S )
t106_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for I being Element of A st I is_terminating_wrt f holds for P being set holds I is_terminating_wrt f,P
t107_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for I being absolutely-terminating Element of A for P being set holds I is_terminating_wrt f,P by Th104, Th106;
t108_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for I being Element of A holds S is_invariant_wrt I,f
t109_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for P being set for I, J being Element of A st P is_invariant_wrt I,f & P is_invariant_wrt J,f holds P is_invariant_wrt I \; J,f
t11_aofa_000:: for f, g being Function for a, A being set st rng f c= dom f & a in dom f & g * f = g & a in A holds ((A,g) iter f) . a = ((A,g) iter f) . (f . a)
t110_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for I, J being Element of A st I is_terminating_wrt f & J is_terminating_wrt f holds I \; J is_terminating_wrt f
t111_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for P being set for I, J being Element of A st I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt I,f holds I \; J is_terminating_wrt f,P
t112_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for C, I, J being Element of A st C is_terminating_wrt f & I is_terminating_wrt f & J is_terminating_wrt f holds if-then-else (C,I,J) is_terminating_wrt f
t113_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for P being set for C, I, J being Element of A st C is_terminating_wrt f,P & I is_terminating_wrt f,P & J is_terminating_wrt f,P & P is_invariant_wrt C,f holds if-then-else (C,I,J) is_terminating_wrt f,P
t114_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f & f iteration_terminates_for I \; C,f . (s,C) holds [s,(while (C,I))] in TerminatingPrograms (A,S,T,f)
t115_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T for P being set for C, I being Element of A st C is_terminating_wrt f,P & I is_terminating_wrt f,P & P is_invariant_wrt C,f & P is_invariant_wrt I,f & f iteration_terminates_for I \; C,f . (s,C) & s in P holds [s,(while (C,I))] in TerminatingPrograms (A,S,T,f)
t116_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T for P being set for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f,P & P is_invariant_wrt C,f & ( for s being Element of S st s in P & f . ((f . (s,I)),C) in T holds f . (s,I) in P ) & f iteration_terminates_for I \; C,f . (s,C) & s in P holds [s,(while (C,I))] in TerminatingPrograms (A,S,T,f)
t117_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f & ( for s being Element of S holds f iteration_terminates_for I \; C,s ) holds while (C,I) is_terminating_wrt f
t118_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for P being set for C, I being Element of A st C is_terminating_wrt f & I is_terminating_wrt f,P & P is_invariant_wrt C,f & ( for s being Element of S st s in P & f . ((f . (s,I)),C) in T holds f . (s,I) in P ) & ( for s being Element of S st f . (s,C) in P holds f iteration_terminates_for I \; C,f . (s,C) ) holds while (C,I) is_terminating_wrt f,P
t12_aofa_000:: for f, g being Function for a, A being set st rng f c= dom f & a in dom f & a nin A holds ((A,g) iter f) . a = a
t13_aofa_000:: for X being non empty set for S being non empty FinSequence of NAT ex A being Universal_Algebra st ( the carrier of A = X & signature A = S )
t14_aofa_000:: for S being non empty FinSequence of NAT ex A being Universal_Algebra st ( the carrier of A = NAT & signature A = S & ( for i, j being Nat st i in dom S & j = S . i holds the charact of A . i = (j -tuples_on NAT) --> i ) )
t15_aofa_000:: for S being non empty FinSequence of NAT for i, j being Nat st i in dom S & j = S . i holds for X being non empty set for f being Function of (j -tuples_on X),X ex A being Universal_Algebra st ( the carrier of A = X & signature A = S & the charact of A . i = f )
t16_aofa_000:: for G being non empty DTConstrStr for t being set holds ( not t in TS G or ex d being Symbol of G st ( d in Terminals G & t = root-tree d ) or ex o being Symbol of G ex p being FinSequence of TS G st ( o ==> roots p & t = o -tree p ) )
t17_aofa_000:: for X being non empty disjoint_with_NAT set for S being non empty FinSequence of NAT for i being Nat st i in dom S holds for p being FinSequence of (FreeUnivAlgNSG (S,X)) st len p = S . i holds (Den ((In (i,(dom the charact of (FreeUnivAlgNSG (S,X))))),(FreeUnivAlgNSG (S,X)))) . p = i -tree p
t18_aofa_000:: for A being Universal_Algebra for B being Subset of A holds B |^ 0 = B
t19_aofa_000:: for A being Universal_Algebra for B being Subset of A for n being Nat holds B |^ (n + 1) = (B |^ n) \/ { ((Den (o,A)) . p) where o is Element of dom the charact of A, p is Element of the carrier of A * : ( p in dom (Den (o,A)) & rng p c= B |^ n ) }
t2_aofa_000:: for X being non empty set for x being Element of X for n being Nat holds arity ((n -tuples_on X) --> x) = n
t20_aofa_000:: for A being Universal_Algebra for B being Subset of A for n being Nat for x being set holds ( x in B |^ (n + 1) iff ( x in B |^ n or ex o being Element of dom the charact of A ex p being Element of the carrier of A * st ( x = (Den (o,A)) . p & p in dom (Den (o,A)) & rng p c= B |^ n ) ) )
t21_aofa_000:: for A being Universal_Algebra for B being Subset of A for n, m being Nat st n <= m holds B |^ n c= B |^ m
t22_aofa_000:: for A being Universal_Algebra for B1, B2 being Subset of A st B1 c= B2 holds for n being Nat holds B1 |^ n c= B2 |^ n
t23_aofa_000:: for A being Universal_Algebra for B being Subset of A for n being Nat for x being set holds ( x in B |^ (n + 1) iff ( x in B or ex o being Element of dom the charact of A ex p being Element of the carrier of A * st ( x = (Den (o,A)) . p & p in dom (Den (o,A)) & rng p c= B |^ n ) ) )
t24_aofa_000:: for A being Universal_Algebra for B being Subset of A ex C being Subset of A st ( C = union { (B |^ n) where n is Element of NAT : verum } & C is opers_closed )
t25_aofa_000:: for A being Universal_Algebra for B, C being Subset of A st C is opers_closed & B c= C holds union { (B |^ n) where n is Element of NAT : verum } c= C
t26_aofa_000:: for A being Universal_Algebra for a being Element of A holds ( a in Generators A iff for o being Element of Operations A holds not a in rng o )
t27_aofa_000:: for A being Universal_Algebra for B being Subset of A st B is opers_closed holds Constants A c= B
t28_aofa_000:: for A being Universal_Algebra st Constants A = {} holds {} A is opers_closed
t29_aofa_000:: for A being Universal_Algebra st Constants A = {} holds for G being GeneratorSet of A holds G <> {}
t3_aofa_000:: for X being non empty set for p being FinSequence of FinTrees X for x, t being set st t in rng p holds t <> x -tree p
t30_aofa_000:: for A being Universal_Algebra for G being Subset of A holds ( G is GeneratorSet of A iff for I being Element of A ex n being Nat st I in G |^ n )
t31_aofa_000:: for A being Universal_Algebra for B being Subset of A for G being GeneratorSet of A st G c= B holds B is GeneratorSet of A
t32_aofa_000:: for A being Universal_Algebra for G being GeneratorSet of A for a being Element of A st ( for o being Element of Operations A holds not a in rng o ) holds a in G
t33_aofa_000:: for A being Universal_Algebra for G being GeneratorSet of A holds Generators A c= G
t34_aofa_000:: for A being free Universal_Algebra for G being free GeneratorSet of A holds G = Generators A
t35_aofa_000:: for A being free Universal_Algebra for G being GeneratorSet of A for B being Universal_Algebra for h1, h2 being Function of A,B st h1 is_homomorphism A,B & h2 is_homomorphism A,B & h1 | G = h2 | G holds h1 = h2
t36_aofa_000:: for A being free Universal_Algebra for o1, o2 being OperSymbol of A for p1, p2 being FinSequence st p1 in dom (Den (o1,A)) & p2 in dom (Den (o2,A)) & (Den (o1,A)) . p1 = (Den (o2,A)) . p2 holds ( o1 = o2 & p1 = p2 )
t37_aofa_000:: for A being free Universal_Algebra for o1, o2 being Element of Operations A for p1, p2 being FinSequence st p1 in dom o1 & p2 in dom o2 & o1 . p1 = o2 . p2 holds ( o1 = o2 & p1 = p2 )
t38_aofa_000:: for A being free Universal_Algebra for o being OperSymbol of A for p being FinSequence st p in dom (Den (o,A)) holds for a being set st a in rng p holds a <> (Den (o,A)) . p
t39_aofa_000:: for A being free Universal_Algebra for G being GeneratorSet of A for o being OperSymbol of A st ( for o9 being OperSymbol of A for p being FinSequence st p in dom (Den (o9,A)) & (Den (o9,A)) . p in G holds o9 <> o ) holds for p being FinSequence st p in dom (Den (o,A)) holds for n being Nat st (Den (o,A)) . p in G |^ (n + 1) holds rng p c= G |^ n
t4_aofa_000:: for f, g being Function for x, X being set st x in X & X c= dom f holds ((f,X) +* g) . x = f . x
t40_aofa_000:: for A being free Universal_Algebra for o being OperSymbol of A for p being FinSequence st p in dom (Den (o,A)) holds for n being Nat st (Den (o,A)) . p in (Generators A) |^ (n + 1) holds rng p c= (Generators A) |^ n
t41_aofa_000:: for X being non empty set for x being Element of X for c being non empty homogeneous quasi_total 2 -ary associative unital PartFunc of (X *),X st x is_a_unity_wrt c holds for i being non empty homogeneous quasi_total 3 -ary PartFunc of (X *),X for w being non empty homogeneous quasi_total 2 -ary PartFunc of (X *),X ex S being strict non-empty UAStr st ( the carrier of S = X & the charact of S = <*((0 -tuples_on X) --> x),c*> ^ <*i,w*> & S is with_empty-instruction & S is with_catenation & S is unital & S is associative & S is with_if-instruction & S is with_while-instruction & S is quasi_total & S is partial )
t42_aofa_000:: for A being non-empty with_empty-instruction UAStr holds dom (Den ((In (1,(dom the charact of A))),A)) = {{}}
t43_aofa_000:: for A being with_empty-instruction Universal_Algebra for o being Element of Operations A st o = Den ((In (1,(dom the charact of A))),A) holds ( arity o = 0 & EmptyIns A in rng o )
t44_aofa_000:: for A being non-empty with_catenation UAStr holds dom (Den ((In (2,(dom the charact of A))),A)) = 2 -tuples_on the carrier of A
t45_aofa_000:: for A being non-empty with_empty-instruction with_catenation unital UAStr for I being Element of A holds ( (EmptyIns A) \; I = I & I \; (EmptyIns A) = I )
t46_aofa_000:: for A being non-empty with_catenation associative UAStr for I1, I2, I3 being Element of A holds (I1 \; I2) \; I3 = I1 \; (I2 \; I3)
t47_aofa_000:: for A being non-empty with_if-instruction UAStr holds dom (Den ((In (3,(dom the charact of A))),A)) = 3 -tuples_on the carrier of A
t48_aofa_000:: for A being non-empty with_while-instruction UAStr holds dom (Den ((In (4,(dom the charact of A))),A)) = 2 -tuples_on the carrier of A
t49_aofa_000:: for A being preIfWhileAlgebra holds EmptyIns A nin ElementaryInstructions A
t5_aofa_000:: for f, g being Function for x, X being set st x nin X & x in dom g holds ((f,X) +* g) . x = g . x
t50_aofa_000:: for A being preIfWhileAlgebra for I1, I2 being Element of A st I1 <> I1 \; I2 & I2 <> I1 \; I2 holds I1 \; I2 nin ElementaryInstructions A
t51_aofa_000:: for A being preIfWhileAlgebra for C, I1, I2 being Element of A holds if-then-else (C,I1,I2) nin ElementaryInstructions A
t52_aofa_000:: for A being preIfWhileAlgebra for C, I being Element of A holds while (C,I) nin ElementaryInstructions A
t53_aofa_000:: for A being preIfWhileAlgebra for I being Element of A holds ( not I nin ElementaryInstructions A or I = EmptyIns A or ex I1, I2 being Element of A st ( I = I1 \; I2 & I1 <> I1 \; I2 & I2 <> I1 \; I2 ) or ex C, I1, I2 being Element of A st I = if-then-else (C,I1,I2) or ex C, J being Element of A st I = while (C,J) )
t54_aofa_000:: ( len ECIW-signature = 4 & dom ECIW-signature = Seg 4 & ECIW-signature . 1 = 0 & ECIW-signature . 2 = 2 & ECIW-signature . 3 = 3 & ECIW-signature . 4 = 2 )
t55_aofa_000:: for A being non empty partial non-empty UAStr st A is ECIW-strict holds for o being OperSymbol of A holds ( o = 1 or o = 2 or o = 3 or o = 4 )
t56_aofa_000:: for X being non empty disjoint_with_NAT set for I being Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds ( ex x being Element of X st I = root-tree x or ex n being Nat ex p being FinSequence of (FreeUnivAlgNSG (ECIW-signature,X)) st ( n in Seg 4 & I = n -tree p & len p = ECIW-signature . n ) )
t57_aofa_000:: for X being non empty disjoint_with_NAT set holds EmptyIns (FreeUnivAlgNSG (ECIW-signature,X)) = 1 -tree {}
t58_aofa_000:: for X being non empty disjoint_with_NAT set for p being FinSequence of (FreeUnivAlgNSG (ECIW-signature,X)) st 1 -tree p is Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds p = {}
t59_aofa_000:: for X being non empty disjoint_with_NAT set for I1, I2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds I1 \; I2 = 2 -tree (I1,I2)
t6_aofa_000:: for f being Function for x being set st x in dom f holds x in f orbit x
t60_aofa_000:: for X being non empty disjoint_with_NAT set for p being FinSequence of (FreeUnivAlgNSG (ECIW-signature,X)) st 2 -tree p is Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds ex I1, I2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) st p = <*I1,I2*>
t61_aofa_000:: for X being non empty disjoint_with_NAT set for I1, I2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds ( I1 \; I2 <> I1 & I1 \; I2 <> I2 )
t62_aofa_000:: for X being non empty disjoint_with_NAT set for I1, I2, J1, J2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) st I1 \; I2 = J1 \; J2 holds ( I1 = J1 & I2 = J2 )
t63_aofa_000:: for X being non empty disjoint_with_NAT set for C, I1, I2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds if-then-else (C,I1,I2) = 3 -tree <*C,I1,I2*>
t64_aofa_000:: for X being non empty disjoint_with_NAT set for p being FinSequence of (FreeUnivAlgNSG (ECIW-signature,X)) st 3 -tree p is Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds ex C, I1, I2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) st p = <*C,I1,I2*>
t65_aofa_000:: for X being non empty disjoint_with_NAT set for C1, C2, I1, I2, J1, J2 being Element of (FreeUnivAlgNSG (ECIW-signature,X)) st if-then-else (C1,I1,I2) = if-then-else (C2,J1,J2) holds ( C1 = C2 & I1 = J1 & I2 = J2 )
t66_aofa_000:: for X being non empty disjoint_with_NAT set for C, I being Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds while (C,I) = 4 -tree (C,I)
t67_aofa_000:: for X being non empty disjoint_with_NAT set for p being FinSequence of (FreeUnivAlgNSG (ECIW-signature,X)) st 4 -tree p is Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds ex C, I being Element of (FreeUnivAlgNSG (ECIW-signature,X)) st p = <*C,I*>
t68_aofa_000:: for X being non empty disjoint_with_NAT set for I being Element of (FreeUnivAlgNSG (ECIW-signature,X)) st I in ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,X)) holds ex x being Element of X st I = x -tree {}
t69_aofa_000:: for X being non empty disjoint_with_NAT set for p being FinSequence of (FreeUnivAlgNSG (ECIW-signature,X)) for x being Element of X st x -tree p is Element of (FreeUnivAlgNSG (ECIW-signature,X)) holds p = {}
t7_aofa_000:: for f being Function for x, y being set st rng f c= dom f & y in f orbit x holds f . y in f orbit x
t70_aofa_000:: for X being non empty disjoint_with_NAT set holds ( ElementaryInstructions (FreeUnivAlgNSG (ECIW-signature,X)) = FreeGenSetNSG (ECIW-signature,X) & card X = card (FreeGenSetNSG (ECIW-signature,X)) )
t71_aofa_000:: for A being preIfWhileAlgebra holds Generators A c= ElementaryInstructions A
t72_aofa_000:: for A being preIfWhileAlgebra st A is free holds for C, I1, I2 being Element of A holds ( EmptyIns A <> I1 \; I2 & EmptyIns A <> if-then-else (C,I1,I2) & EmptyIns A <> while (C,I1) )
t73_aofa_000:: for A being preIfWhileAlgebra st A is free holds for I1, I2, C, J1, J2 being Element of A holds ( I1 \; I2 <> I1 & I1 \; I2 <> I2 & ( I1 \; I2 = J1 \; J2 implies ( I1 = J1 & I2 = J2 ) ) & I1 \; I2 <> if-then-else (C,J1,J2) & I1 \; I2 <> while (C,J1) )
t74_aofa_000:: for A being preIfWhileAlgebra st A is free holds for C, I1, I2, D, J1, J2 being Element of A holds ( if-then-else (C,I1,I2) <> C & if-then-else (C,I1,I2) <> I1 & if-then-else (C,I1,I2) <> I2 & if-then-else (C,I1,I2) <> while (D,J1) & ( if-then-else (C,I1,I2) = if-then-else (D,J1,J2) implies ( C = D & I1 = J1 & I2 = J2 ) ) )
t75_aofa_000:: for A being preIfWhileAlgebra st A is free holds for C, I, D, J being Element of A holds ( while (C,I) <> C & while (C,I) <> I & ( while (C,I) = while (D,J) implies ( C = D & I = J ) ) )
t76_aofa_000:: for A being preIfWhileAlgebra for B being Subset of A for n being Nat holds ( EmptyIns A in B |^ (n + 1) & ( for C, I1, I2 being Element of A st C in B |^ n & I1 in B |^ n & I2 in B |^ n holds ( I1 \; I2 in B |^ (n + 1) & if-then-else (C,I1,I2) in B |^ (n + 1) & while (C,I1) in B |^ (n + 1) ) ) )
t77_aofa_000:: for A being ECIW-strict preIfWhileAlgebra for x being set for n being Nat holds ( not x in (ElementaryInstructions A) |^ (n + 1) or x in (ElementaryInstructions A) |^ n or x = EmptyIns A or ex I1, I2 being Element of A st ( x = I1 \; I2 & I1 in (ElementaryInstructions A) |^ n & I2 in (ElementaryInstructions A) |^ n ) or ex C, I1, I2 being Element of A st ( x = if-then-else (C,I1,I2) & C in (ElementaryInstructions A) |^ n & I1 in (ElementaryInstructions A) |^ n & I2 in (ElementaryInstructions A) |^ n ) or ex C, I being Element of A st ( x = while (C,I) & C in (ElementaryInstructions A) |^ n & I in (ElementaryInstructions A) |^ n ) )
t78_aofa_000:: for A being Universal_Algebra for B being Subset of A holds Constants A c= B |^ 1
t79_aofa_000:: for A being preIfWhileAlgebra holds ( A is well_founded iff for I being Element of A ex n being Nat st I in (ElementaryInstructions A) |^ n )
t8_aofa_000:: for f being Function for x being set st x in dom f holds f . x in f orbit x
t80_aofa_000:: for A being preIfWhileAlgebra for C, I being Element of A for S being non empty set for T being Subset of S for f being Function of [:S, the carrier of A:],S st f is complying_with_empty-instruction & f complies_with_if_wrt T holds for s being Element of S st f . (s,C) nin T holds f . (s,(if-then (C,I))) = f . (s,C)
t81_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S holds ( pr1 (S, the carrier of A) is complying_with_empty-instruction & pr1 (S, the carrier of A) is complying_with_catenation & pr1 (S, the carrier of A) complies_with_if_wrt T & pr1 (S, the carrier of A) complies_with_while_wrt T )
t82_aofa_000:: for A being preIfWhileAlgebra for I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T holds ( f iteration_terminates_for I,s iff iteration-degree (I,s,f) < +infty )
t83_aofa_000:: for A being preIfWhileAlgebra for I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T st s nin T holds ( f iteration_terminates_for I,s & iteration-degree (I,s,f) = 0 )
t84_aofa_000:: for A being preIfWhileAlgebra for I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T st s in T holds ( ( f iteration_terminates_for I,s implies f iteration_terminates_for I,f . (s,I) ) & ( f iteration_terminates_for I,f . (s,I) implies f iteration_terminates_for I,s ) & iteration-degree (I,s,f) = 1. + (iteration-degree (I,(f . (s,I)),f)) )
t85_aofa_000:: for A being preIfWhileAlgebra for I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T holds iteration-degree (I,s,f) >= 0
t86_aofa_000:: for A being preIfWhileAlgebra for C, I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T for r being non empty FinSequence of S st r . 1 = f . (s,C) & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds ( r . i in T & r . (i + 1) = f . ((r . i),(I \; C)) ) ) holds f . (s,(while (C,I))) = r . (len r)
t87_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for f being ExecutionFunction of A,S,T for I being Element of A for s being Element of S holds ( not f iteration_terminates_for I,s iff ((curry' f) . I) orbit s c= T )
t88_aofa_000:: for A being free preIfWhileAlgebra for I1, I2 being Element of A for n being Nat st I1 \; I2 in (ElementaryInstructions A) |^ n holds ex i being Nat st ( n = i + 1 & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
t89_aofa_000:: for A being free preIfWhileAlgebra for C, I1, I2 being Element of A for n being Nat st if-then-else (C,I1,I2) in (ElementaryInstructions A) |^ n holds ex i being Nat st ( n = i + 1 & C in (ElementaryInstructions A) |^ i & I1 in (ElementaryInstructions A) |^ i & I2 in (ElementaryInstructions A) |^ i )
t9_aofa_000:: for f being Function for x being set st x in dom f holds f orbit (f . x) c= f orbit x
t90_aofa_000:: for A being free preIfWhileAlgebra for C, I being Element of A for n being Nat st while (C,I) in (ElementaryInstructions A) |^ n holds ex i being Nat st ( n = i + 1 & C in (ElementaryInstructions A) |^ i & I in (ElementaryInstructions A) |^ i )
t91_aofa_000:: for S being non empty set for T being Subset of S for A being free ECIW-strict preIfWhileAlgebra for g being Function of [:S,(ElementaryInstructions A):],S for s0 being Element of S ex f being ExecutionFunction of A,S,T st ( f | [:S,(ElementaryInstructions A):] = g & ( for s being Element of S for C, I being Element of A st not f iteration_terminates_for I \; C,f . (s,C) holds f . (s,(while (C,I))) = s0 ) )
t92_aofa_000:: for S being non empty set for T being Subset of S for A being free ECIW-strict preIfWhileAlgebra for g being Function of [:S,(ElementaryInstructions A):],S for F being Function of (Funcs (S,S)),(Funcs (S,S)) st ( for h being Element of Funcs (S,S) holds (F . h) * h = F . h ) holds ex f being ExecutionFunction of A,S,T st ( f | [:S,(ElementaryInstructions A):] = g & ( for C, I being Element of A for s being Element of S st not f iteration_terminates_for I \; C,f . (s,C) holds f . (s,(while (C,I))) = (F . ((curry' f) . (I \; C))) . (f . (s,C)) ) )
t93_aofa_000:: for S being non empty set for T being Subset of S for A being free ECIW-strict preIfWhileAlgebra for f1, f2 being ExecutionFunction of A,S,T st f1 | [:S,(ElementaryInstructions A):] = f2 | [:S,(ElementaryInstructions A):] & ( for s being Element of S for C, I being Element of A st not f1 iteration_terminates_for I \; C,f1 . (s,C) holds f1 . (s,(while (C,I))) = f2 . (s,(while (C,I))) ) holds f1 = f2
t94_aofa_000:: for A being preIfWhileAlgebra for I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T st I in ElementaryInstructions A holds [s,I] in TerminatingPrograms (A,S,T,f)
t95_aofa_000:: for A being preIfWhileAlgebra for I being Element of A st I in ElementaryInstructions A holds I is absolutely-terminating
t96_aofa_000:: for A being preIfWhileAlgebra for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T holds [s,(EmptyIns A)] in TerminatingPrograms (A,S,T,f)
t97_aofa_000:: for A being preIfWhileAlgebra for I, J being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T st A is free & [s,(I \; J)] in TerminatingPrograms (A,S,T,f) holds ( [s,I] in TerminatingPrograms (A,S,T,f) & [(f . (s,I)),J] in TerminatingPrograms (A,S,T,f) )
t98_aofa_000:: for A being preIfWhileAlgebra for C, I, J being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T st A is free & [s,(if-then-else (C,I,J))] in TerminatingPrograms (A,S,T,f) holds ( [s,C] in TerminatingPrograms (A,S,T,f) & ( f . (s,C) in T implies [(f . (s,C)),I] in TerminatingPrograms (A,S,T,f) ) & ( f . (s,C) nin T implies [(f . (s,C)),J] in TerminatingPrograms (A,S,T,f) ) )
t99_aofa_000:: for A being preIfWhileAlgebra for C, I being Element of A for S being non empty set for T being Subset of S for s being Element of S for f being ExecutionFunction of A,S,T st A is free & [s,(while (C,I))] in TerminatingPrograms (A,S,T,f) holds ( [s,C] in TerminatingPrograms (A,S,T,f) & ex r being non empty FinSequence of S st ( r . 1 = f . (s,C) & r . (len r) nin T & ( for i being Nat st 1 <= i & i < len r holds ( r . i in T & [(r . i),(I \; C)] in TerminatingPrograms (A,S,T,f) & r . (i + 1) = f . ((r . i),(I \; C)) ) ) ) )
d1_aofa_i00:: for F being non empty functional set for x, f being set holds F \ (x,f) = { g where g is Element of F : g . x <> f } ;
d10_aofa_i00:: for X being non empty set for f being Function of X,INT for x being integer number for b4 being Function of X,INT holds ( b4 = f * x iff for s being Element of X holds b4 . s = (f . s) * x );
d11_aofa_i00:: for X being non empty set for t1, t2, b4 being Function of X,INT holds ( b4 = t1 - t2 iff for s being Element of X holds b4 . s = (t1 . s) - (t2 . s) );
d12_aofa_i00:: for X being non empty set for t1, t2, b4 being Function of X,INT holds ( b4 = t1 + t2 iff for s being Element of X holds b4 . s = (t1 . s) + (t2 . s) );
d13_aofa_i00:: for N being set for v, f, b4 being Function holds ( b4 = v ** (f,N) iff ( ex Y being set st ( ( for y being set holds ( y in Y iff ex h being Function st ( h in dom v & y in rng h ) ) ) & ( for a being set holds ( a in dom b4 iff ( a in Funcs (N,Y) & ex g being Function st ( a = g & g * f in dom v ) ) ) ) ) & ( for g being Function st g in dom b4 holds b4 . g = v . (g * f) ) ) );
d14_aofa_i00:: for A being preIfWhileAlgebra for I being Element of A for X being non empty set for T being Subset of (Funcs (X,INT)) for f being ExecutionFunction of A, Funcs (X,INT),T holds ( I is_assignment_wrt A,X,f iff ( I in ElementaryInstructions A & ex v being INT-Variable of X ex t being INT-Expression of X st for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) );
d15_aofa_i00:: for A being preIfWhileAlgebra for X being non empty set for T being Subset of (Funcs (X,INT)) for f being ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of X for t being INT-Expression of X holds ( v,t form_assignment_wrt f iff ex I being Element of A st ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) );
d16_aofa_i00:: for A being preIfWhileAlgebra for X being non empty set for T being Subset of (Funcs (X,INT)) for f being ExecutionFunction of A, Funcs (X,INT),T st ex I being Element of A st I is_assignment_wrt A,X,f holds for b5 being INT-Variable of X holds ( b5 is INT-Variable of A,f iff ex t being INT-Expression of X st b5,t form_assignment_wrt f );
d17_aofa_i00:: for A being preIfWhileAlgebra for X being non empty set for T being Subset of (Funcs (X,INT)) for f being ExecutionFunction of A, Funcs (X,INT),T st ex I being Element of A st I is_assignment_wrt A,X,f holds for b5 being INT-Expression of X holds ( b5 is INT-Expression of A,f iff ex v being INT-Variable of X st v,b5 form_assignment_wrt f );
d18_aofa_i00:: for X being non empty set for x being Element of X for b3 being INT-Expression of X holds ( b3 = . x iff for s being Element of Funcs (X,INT) holds b3 . s = s . x );
d19_aofa_i00:: for X being non empty set for v being INT-Variable of X for b3 being INT-Expression of X holds ( b3 = . v iff for s being Element of Funcs (X,INT) holds b3 . s = s . (v . s) );
d2_aofa_i00:: for X, Y, Z being set for f being Function of [:(Funcs (X,INT)),Y:],Z for b5 being Element of X holds ( b5 is Variable of f iff verum );
d20_aofa_i00:: for X being non empty set for x being Element of X holds ^ x = (Funcs (X,INT)) --> x;
d21_aofa_i00:: for X being non empty set for i being integer number holds . (i,X) = (Funcs (X,INT)) --> i;
d22_aofa_i00:: for A being preIfWhileAlgebra for X being non empty set for T being Subset of (Funcs (X,INT)) for f being ExecutionFunction of A, Funcs (X,INT),T holds ( f is Euclidean iff ( ( for v being INT-Variable of A,f for t being INT-Expression of A,f holds v,t form_assignment_wrt f ) & ( for i being integer number holds . (i,X) is INT-Expression of A,f ) & ( for v being INT-Variable of A,f holds . v is INT-Expression of A,f ) & ( for x being Element of X holds ^ x is INT-Variable of A,f ) & ex a being INT-Array of X st ( a | (card X) is one-to-one & ( for t being INT-Expression of A,f holds a * t is INT-Variable of A,f ) ) & ( for t being INT-Expression of A,f holds - t is INT-Expression of A,f ) & ( for t1, t2 being INT-Expression of A,f holds ( t1 (#) t2 is INT-Expression of A,f & t1 + t2 is INT-Expression of A,f & t1 div t2 is INT-Expression of A,f & t1 mod t2 is INT-Expression of A,f & leq (t1,t2) is INT-Expression of A,f & gt (t1,t2) is INT-Expression of A,f ) ) ) );
d23_aofa_i00:: for A being preIfWhileAlgebra holds ( A is Euclidean iff for X being non empty countable set for T being Subset of (Funcs (X,INT)) ex f being ExecutionFunction of A, Funcs (X,INT),T st f is Euclidean );
d24_aofa_i00:: INT-ElemIns = [:(Funcs ((Funcs (NAT,INT)),NAT)),(Funcs ((Funcs (NAT,INT)),INT)):];
d25_aofa_i00:: for b1 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (NAT,INT),(Funcs (NAT,INT)) \ (0,0) holds ( b1 is INT-Exec iff for s being Element of Funcs (NAT,INT) for v being Element of Funcs ((Funcs (NAT,INT)),NAT) for e being Element of Funcs ((Funcs (NAT,INT)),INT) holds b1 . (s,(root-tree [v,e])) = s +* ((v . s),(e . s)) );
d26_aofa_i00:: for X being non empty set holds INT-ElemIns X = [:(Funcs ((Funcs (X,INT)),X)),(Funcs ((Funcs (X,INT)),INT)):];
d27_aofa_i00:: for X being non empty set for x being Element of X for b3 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,(INT-ElemIns X)), Funcs (X,INT),(Funcs (X,INT)) \ (x,0) holds ( b3 is INT-Exec of x iff for s being Element of Funcs (X,INT) for v being Element of Funcs ((Funcs (X,INT)),X) for e being Element of Funcs ((Funcs (X,INT)),INT) holds b3 . (s,(root-tree [v,e])) = s +* ((v . s),(e . s)) );
d28_aofa_i00:: for X being non empty set for T being Subset of (Funcs (X,INT)) for c being Enumeration of X st rng c c= NAT holds for b4 being ExecutionFunction of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns), Funcs (X,INT),T holds ( b4 is INT-Exec of c,T iff for s being Element of Funcs (X,INT) for v being Element of Funcs ((Funcs (X,INT)),X) for e being Element of Funcs ((Funcs (X,INT)),INT) holds b4 . (s,(root-tree [((c * v) ** (c,NAT)),(e ** (c,NAT))])) = s +* ((v . s),(e . s)) );
d29_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for t1, t2, b7 being INT-Expression of A,f holds ( b7 = t1 div t2 iff for s being Element of Funcs (X,INT) holds b7 . s = (t1 . s) div (t2 . s) );
d3_aofa_i00:: for t1, t2, b3 being INT -valued Function holds ( b3 = t1 div t2 iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds b3 . s = (t1 . s) div (t2 . s) ) ) );
d30_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for t1, t2, b7 being INT-Expression of A,f holds ( b7 = t1 mod t2 iff for s being Element of Funcs (X,INT) holds b7 . s = (t1 . s) mod (t2 . s) );
d31_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for t1, t2, b7 being INT-Expression of A,f holds ( b7 = leq (t1,t2) iff for s being Element of Funcs (X,INT) holds b7 . s = IFGT ((t1 . s),(t2 . s),0,1) );
d32_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for t1, t2, b7 being INT-Expression of A,f holds ( b7 = gt (t1,t2) iff for s being Element of Funcs (X,INT) holds b7 . s = IFGT ((t1 . s),(t2 . s),1,0) );
d33_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for t1, t2, b7 being INT-Expression of A,f holds ( b7 = eq (t1,t2) iff for s being Element of Funcs (X,INT) holds b7 . s = IFEQ ((t1 . s),(t2 . s),1,0) );
d34_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f holds . v = . v;
d35_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Element of X holds x ^ (A,f) = ^ x;
d36_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f holds . x = . (^ x);
d37_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for i being integer number holds . (i,A,f) = . (i,X);
d38_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for t being INT-Expression of A,f holds v := t = choose { I where I is Element of A : ( I in ElementaryInstructions A & ( for s being Element of Funcs (X,INT) holds f . (s,I) = s +* ((v . s),(t . s)) ) ) } ;
d39_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for t being INT-Expression of A,f holds v += t = v := ((. v) + t);
d4_aofa_i00:: for t1, t2, b3 being INT -valued Function holds ( b3 = t1 mod t2 iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds b3 . s = (t1 . s) mod (t2 . s) ) ) );
d40_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for t being INT-Expression of A,f holds v *= t = v := ((. v) (#) t);
d41_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Element of X for t being INT-Expression of A,f holds x := t = (x ^ (A,f)) := t;
d42_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Element of X for y being Variable of f holds x := y = x := (. y);
d43_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Element of X for v being INT-Variable of A,f holds x := v = x := (. v);
d44_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v, w being INT-Variable of A,f holds v := w = v := (. w);
d45_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for i being integer number holds x := i = x := (. (i,A,f));
d46_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v1, v2 being INT-Variable of A,f for x being Variable of f holds swap (v1,x,v2) = ((x := v1) \; (v1 := v2)) \; (v2 := (. x));
d47_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds x += t = x := ((. x) + t);
d48_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds x *= t = x := ((. x) (#) t);
d49_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds x %= t = x := ((. x) mod t);
d5_aofa_i00:: for t1, t2, b3 being INT -valued Function holds ( b3 = leq (t1,t2) iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds b3 . s = IFGT ((t1 . s),(t2 . s),0,1) ) ) );
d50_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds x /= t = x := ((. x) div t);
d51_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for i being integer number holds x += i = x := ((. x) + i);
d52_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for i being integer number holds x *= i = x := ((. x) * i);
d53_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for i being integer number holds x %= i = x := ((. x) mod (. (i,A,f)));
d54_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for i being integer number holds x /= i = x := ((. x) div (. (i,A,f)));
d55_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for i being integer number holds x div i = (. x) div (. (i,A,f));
d56_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for i being integer number holds v := i = v := (. (i,A,f));
d57_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for i being integer number holds v += i = v := ((. v) + i);
d58_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for i being integer number holds v *= i = v := ((. v) * i);
d59_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for t1 being INT-Expression of A,g holds t1 is_odd = b := (t1 mod (. (2,A,g)));
d6_aofa_i00:: for t1, t2, b3 being INT -valued Function holds ( b3 = gt (t1,t2) iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds b3 . s = IFGT ((t1 . s),(t2 . s),1,0) ) ) );
d60_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for t1 being INT-Expression of A,g holds t1 is_even = b := ((t1 + 1) mod (. (2,A,g)));
d61_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for t1, t2 being INT-Expression of A,g holds t1 leq t2 = b := (leq (t1,t2));
d62_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for t1, t2 being INT-Expression of A,g holds t1 gt t2 = b := (gt (t1,t2));
d63_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for t1, t2 being INT-Expression of A,g holds t1 eq t2 = b := (eq (t1,t2));
d64_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for v1, v2 being INT-Variable of A,g holds v1 leq v2 = (. v1) leq (. v2);
d65_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for v1, v2 being INT-Variable of A,g holds v1 gt v2 = (. v1) gt (. v2);
d66_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x1 being Variable of g holds x1 is_odd = (. x1) is_odd ;
d67_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x1 being Variable of g holds x1 is_even = (. x1) is_even ;
d68_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x1, x2 being Variable of g holds x1 leq x2 = (. x1) leq (. x2);
d69_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x1, x2 being Variable of g holds x1 gt x2 = (. x1) gt (. x2);
d7_aofa_i00:: for t1, t2, b3 being INT -valued Function holds ( b3 = eq (t1,t2) iff ( dom b3 = (dom t1) /\ (dom t2) & ( for s being set st s in dom b3 holds b3 . s = IFEQ ((t1 . s),(t2 . s),1,0) ) ) );
d70_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds x leq i = (. x) leq (. (i,A,g));
d71_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds x geq i = (. x) geq (. (i,A,g));
d72_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds x gt i = (. x) gt (. (i,A,g));
d73_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds x lt i = (. x) lt (. (i,A,g));
d74_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds x / i = (. x) div (. (i,A,g));
d75_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 += x2 = x1 += (. x2);
d76_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 *= x2 = x1 *= (. x2);
d77_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 %= x2 = x1 := ((. x1) mod (. x2));
d78_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 /= x2 = x1 := ((. x1) div (. x2));
d79_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 + x2 = (. x1) + (. x2);
d8_aofa_i00:: for X being non empty set for f being Function of X,INT for x being integer number for b4 being Function of X,INT holds ( b4 = f + x iff for s being Element of X holds b4 . s = (f . s) + x );
d80_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 * x2 = (. x1) (#) (. x2);
d81_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 mod x2 = (. x1) mod (. x2);
d82_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x1, x2 being Variable of f holds x1 div x2 = (. x1) div (. x2);
d9_aofa_i00:: for X being non empty set for f being Function of X,INT for x being integer number for b4 being Function of X,INT holds ( b4 = f - x iff for s being Element of X holds b4 . s = (f . s) - x );
s1_aofa_i00:: scheme ForToIteration{ F1() -> Euclidean preIfWhileAlgebra, F2() -> non empty countable set , F3() -> Element of F2(), F4() -> Element of F1(), F5() -> Element of F1(), F6() -> Euclidean ExecutionFunction of F1(), Funcs (F2(),INT),(Funcs (F2(),INT)) \ (F3(),0), F7() -> Variable of F6(), F8() -> Variable of F6(), F9() -> Element of Funcs (F2(),INT), F10() -> INT-Expression of F1(),F6(), P1[ set ] } : ( P1[F6() . (F9(),F5())] & ( F10() . F9() <= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) + 1 ) & ( F10() . F9() > F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() ) provided A1: F5() = for-do ((F7() := F10()),(F7() leq F8()),(F7() += 1),F4()) and A2: P1[F6() . (F9(),(F7() := F10()))] and A3: for s being Element of Funcs (F2(),INT) st P1[s] holds ( P1[F6() . (s,(F4() \; (F7() += 1)))] & P1[F6() . (s,(F7() leq F8()))] ) and A4: for s being Element of Funcs (F2(),INT) st P1[s] holds ( (F6() . (s,F4())) . F7() = s . F7() & (F6() . (s,F4())) . F8() = s . F8() ) and A5: ( F8() <> F7() & F8() <> F3() & F7() <> F3() )
s2_aofa_i00:: scheme ForDowntoIteration{ F1() -> Euclidean preIfWhileAlgebra, F2() -> non empty countable set , F3() -> Element of F2(), F4() -> Element of F1(), F5() -> Element of F1(), F6() -> Euclidean ExecutionFunction of F1(), Funcs (F2(),INT),(Funcs (F2(),INT)) \ (F3(),0), F7() -> Variable of F6(), F8() -> Variable of F6(), F9() -> Element of Funcs (F2(),INT), F10() -> INT-Expression of F1(),F6(), P1[ set ] } : ( P1[F6() . (F9(),F5())] & ( F10() . F9() >= F9() . F8() implies (F6() . (F9(),F5())) . F7() = (F9() . F8()) - 1 ) & ( F10() . F9() < F9() . F8() implies (F6() . (F9(),F5())) . F7() = F10() . F9() ) & (F6() . (F9(),F5())) . F8() = F9() . F8() ) provided A1: F5() = for-do ((F7() := F10()),((. F8()) leq (. F7())),(F7() += (- 1)),F4()) and A2: P1[F6() . (F9(),(F7() := F10()))] and A3: for s being Element of Funcs (F2(),INT) st P1[s] holds ( P1[F6() . (s,(F4() \; (F7() += (- 1))))] & P1[F6() . (s,(F8() leq F7()))] ) and A4: for s being Element of Funcs (F2(),INT) st P1[s] holds ( (F6() . (s,F4())) . F7() = s . F7() & (F6() . (s,F4())) . F8() = s . F8() ) and A5: ( F8() <> F7() & F8() <> F3() & F7() <> F3() )
t1_aofa_i00:: for x, y, z, a, b, c being set st a <> b & b <> c & c <> a holds ex f being Function st ( f . a = x & f . b = y & f . c = z )
t10_aofa_i00:: for X being non empty set for f being Denumeration of X holds f . 0 in X by FUNCT_2:5, ORDINAL3:8;
t11_aofa_i00:: for X being countable set for f being Enumeration of X holds rng f c= NAT
t12_aofa_i00:: for n, m being Nat holds 0 to_power (n + m) = (0 to_power n) * (0 to_power m)
t13_aofa_i00:: for x being real number for n, m being Nat holds (x to_power n) to_power m = x to_power (n * m) by NEWTON:9;
t14_aofa_i00:: for X being non empty set for x being Element of X holds . x = . (^ x)
t15_aofa_i00:: for X being non empty set for t being INT-Expression of X holds ( t + (. (0,X)) = t & t (#) (. (1,X)) = t )
t16_aofa_i00:: for f being INT-Exec for v being INT-Variable of NAT for t being INT-Expression of NAT holds v,t form_assignment_wrt f
t17_aofa_i00:: for f being INT-Exec for v being INT-Variable of NAT holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
t18_aofa_i00:: for f being INT-Exec for t being INT-Expression of NAT holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
t19_aofa_i00:: for X being non empty countable set for T being Subset of (Funcs (X,INT)) for c being Enumeration of X for f being INT-Exec of c,T for v being INT-Variable of X for t being INT-Expression of X holds v,t form_assignment_wrt f
t2_aofa_i00:: for F being non empty functional set for x, y being set for g being Element of F holds ( g in F \ (x,y) iff g . x <> y )
t20_aofa_i00:: for X being non empty countable set for T being Subset of (Funcs (X,INT)) for c being Enumeration of X for f being INT-Exec of c,T for v being INT-Variable of X holds v is INT-Variable of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
t21_aofa_i00:: for X being non empty countable set for T being Subset of (Funcs (X,INT)) for c being Enumeration of X for f being INT-Exec of c,T for t being INT-Expression of X holds t is INT-Expression of FreeUnivAlgNSG (ECIW-signature,INT-ElemIns),f
t22_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for s being Element of Funcs (X,INT) holds (. x) . s = s . x
t23_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for t being INT-Expression of A,f holds v := t in ElementaryInstructions A
t24_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for v being INT-Variable of A,f for t being INT-Expression of A,f holds ( (f . (s,(v := t))) . (v . s) = t . s & ( for z being Element of X st z <> v . s holds (f . (s,(v := t))) . z = s . z ) )
t25_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for i being integer number holds ( (f . (s,(x := i))) . x = i & ( for z being Element of X st z <> x holds (f . (s,(x := i))) . z = s . z ) )
t26_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds ( (f . (s,(x := t))) . x = t . s & ( for z being Element of X st z <> x holds (f . (s,(x := t))) . z = s . z ) )
t27_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x, y being Variable of f holds ( (f . (s,(x := y))) . x = s . y & ( for z being Element of X st z <> x holds (f . (s,(x := y))) . z = s . z ) )
t28_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for i being integer number for x being Variable of f holds ( (f . (s,(x += i))) . x = (s . x) + i & ( for z being Element of X st z <> x holds (f . (s,(x += i))) . z = s . z ) )
t29_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds ( (f . (s,(x += t))) . x = (s . x) + (t . s) & ( for z being Element of X st z <> x holds (f . (s,(x += t))) . z = s . z ) )
t3_aofa_i00:: for f1, f2, g being Function st rng g c= dom f2 holds (f1 +* f2) * g = f2 * g
t30_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x, y being Variable of f holds ( (f . (s,(x += y))) . x = (s . x) + (s . y) & ( for z being Element of X st z <> x holds (f . (s,(x += y))) . z = s . z ) )
t31_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for i being integer number for x being Variable of f holds ( (f . (s,(x *= i))) . x = (s . x) * i & ( for z being Element of X st z <> x holds (f . (s,(x *= i))) . z = s . z ) )
t32_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds ( (f . (s,(x *= t))) . x = (s . x) * (t . s) & ( for z being Element of X st z <> x holds (f . (s,(x *= t))) . z = s . z ) )
t33_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x, y being Variable of f holds ( (f . (s,(x *= y))) . x = (s . x) * (s . y) & ( for z being Element of X st z <> x holds (f . (s,(x *= y))) . z = s . z ) )
t34_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds ( ( s . x <= i implies (g . (s,(x leq i))) . b = 1 ) & ( s . x > i implies (g . (s,(x leq i))) . b = 0 ) & ( s . x >= i implies (g . (s,(x geq i))) . b = 1 ) & ( s . x < i implies (g . (s,(x geq i))) . b = 0 ) & ( for z being Element of X st z <> b holds ( (g . (s,(x leq i))) . z = s . z & (g . (s,(x geq i))) . z = s . z ) ) )
t35_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y being Variable of g holds ( ( s . x <= s . y implies (g . (s,(x leq y))) . b = 1 ) & ( s . x > s . y implies (g . (s,(x leq y))) . b = 0 ) & ( for z being Element of X st z <> b holds (g . (s,(x leq y))) . z = s . z ) )
t36_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds ( ( s . x <= i implies g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq i)) in (Funcs (X,INT)) \ (b,0) implies s . x <= i ) & ( s . x >= i implies g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq i)) in (Funcs (X,INT)) \ (b,0) implies s . x >= i ) )
t37_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y being Variable of g holds ( ( s . x <= s . y implies g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x leq y)) in (Funcs (X,INT)) \ (b,0) implies s . x <= s . y ) & ( s . x >= s . y implies g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x geq y)) in (Funcs (X,INT)) \ (b,0) implies s . x >= s . y ) )
t38_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds ( ( s . x > i implies (g . (s,(x gt i))) . b = 1 ) & ( s . x <= i implies (g . (s,(x gt i))) . b = 0 ) & ( s . x < i implies (g . (s,(x lt i))) . b = 1 ) & ( s . x >= i implies (g . (s,(x lt i))) . b = 0 ) & ( for z being Element of X st z <> b holds ( (g . (s,(x gt i))) . z = s . z & (g . (s,(x lt i))) . z = s . z ) ) )
t39_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y being Variable of g holds ( ( s . x > s . y implies (g . (s,(x gt y))) . b = 1 ) & ( s . x <= s . y implies (g . (s,(x gt y))) . b = 0 ) & ( s . x < s . y implies (g . (s,(x lt y))) . b = 1 ) & ( s . x >= s . y implies (g . (s,(x lt y))) . b = 0 ) & ( for z being Element of X st z <> b holds ( (g . (s,(x gt y))) . z = s . z & (g . (s,(x lt y))) . z = s . z ) ) )
t4_aofa_i00:: for X, N, I being non empty set for s being Function of X,I for c being Function of X,N st c is one-to-one holds for n being Element of I holds (N --> n) +* (s * (c ")) is Function of N,I
t40_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g for i being integer number holds ( ( s . x > i implies g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt i)) in (Funcs (X,INT)) \ (b,0) implies s . x > i ) & ( s . x < i implies g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt i)) in (Funcs (X,INT)) \ (b,0) implies s . x < i ) )
t41_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y being Variable of g holds ( ( s . x > s . y implies g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x gt y)) in (Funcs (X,INT)) \ (b,0) implies s . x > s . y ) & ( s . x < s . y implies g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x lt y)) in (Funcs (X,INT)) \ (b,0) implies s . x < s . y ) )
t42_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for i being integer number for x being Variable of f holds ( (f . (s,(x %= i))) . x = (s . x) mod i & ( for z being Element of X st z <> x holds (f . (s,(x %= i))) . z = s . z ) )
t43_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds ( (f . (s,(x %= t))) . x = (s . x) mod (t . s) & ( for z being Element of X st z <> x holds (f . (s,(x %= t))) . z = s . z ) )
t44_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x, y being Variable of f holds ( (f . (s,(x %= y))) . x = (s . x) mod (s . y) & ( for z being Element of X st z <> x holds (f . (s,(x %= y))) . z = s . z ) )
t45_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for i being integer number for x being Variable of f holds ( (f . (s,(x /= i))) . x = (s . x) div i & ( for z being Element of X st z <> x holds (f . (s,(x /= i))) . z = s . z ) )
t46_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x being Variable of f for t being INT-Expression of A,f holds ( (f . (s,(x /= t))) . x = (s . x) div (t . s) & ( for z being Element of X st z <> x holds (f . (s,(x /= t))) . z = s . z ) )
t47_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for x, y being Variable of f holds ( (f . (s,(x /= y))) . x = (s . x) div (s . y) & ( for z being Element of X st z <> x holds (f . (s,(x /= y))) . z = s . z ) )
t48_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for t being INT-Expression of A,g holds ( (g . (s,(t is_odd))) . b = (t . s) mod 2 & (g . (s,(t is_even))) . b = ((t . s) + 1) mod 2 & ( for z being Element of X st z <> b holds ( (g . (s,(t is_odd))) . z = s . z & (g . (s,(t is_even))) . z = s . z ) ) )
t49_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g holds ( (g . (s,(x is_odd))) . b = (s . x) mod 2 & (g . (s,(x is_even))) . b = ((s . x) + 1) mod 2 & ( for z being Element of X st z <> b holds (g . (s,(x is_odd))) . z = s . z ) )
t5_aofa_i00:: for N, X, I being non empty set for v1, v2 being Function st dom v1 = dom v2 & dom v1 = Funcs (X,I) holds for f being Function of X,N st f is one-to-one & v1 ** (f,N) = v2 ** (f,N) holds v1 = v2
t50_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for t being INT-Expression of A,g holds ( ( t . s is odd implies g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_odd)) in (Funcs (X,INT)) \ (b,0) implies t . s is odd ) & ( t . s is even implies g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(t is_even)) in (Funcs (X,INT)) \ (b,0) implies t . s is even ) )
t51_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x being Variable of g holds ( ( s . x is odd implies g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_odd)) in (Funcs (X,INT)) \ (b,0) implies s . x is odd ) & ( s . x is even implies g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(x is_even)) in (Funcs (X,INT)) \ (b,0) implies s . x is even ) )
t52_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for I being Element of A for i, n being Variable of g st ex d being Function st ( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
t53_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for s being Element of Funcs (X,INT) for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for P being set for I being Element of A for i, n being Variable of g st ex d being Function st ( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
t54_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for t being INT-Expression of A,f for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for I being Element of A st I is_terminating_wrt g holds for i, n being Variable of g st ex d being Function st ( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g
t55_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for T being Subset of (Funcs (X,INT)) for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T for t being INT-Expression of A,f for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for P being set for I being Element of A st I is_terminating_wrt g,P holds for i, n being Variable of g st ex d being Function st ( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) st s in P holds ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
t56_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for n, s, i being Variable of g st ex d being Function st ( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds (s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i))) is_terminating_wrt g
t57_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for n, s, i being Variable of g st ex d being Function st ( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds for q being Element of Funcs (X,INT) for N being Nat st N = q . n holds (g . (q,((s := 1) \; (for-do ((i := 2),(i leq n),(i += 1),(s *= i)))))) . s = N !
t58_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, n, s, i being Variable of g st ex d being Function st ( d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds (s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x))) is_terminating_wrt g
t59_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, n, s, i being Variable of g st ex d being Function st ( d . x = 0 & d . n = 1 & d . s = 2 & d . i = 3 & d . b = 4 ) holds for q being Element of Funcs (X,INT) for N being Nat st N = q . n holds (g . (q,((s := 1) \; (for-do ((i := 1),(i leq n),(i += 1),(s *= x)))))) . s = (q . x) |^ N
t6_aofa_i00:: for X being set for f being Function holds ( f is Enumeration of X iff ( dom f = X & rng f = card X & f is one-to-one ) )
t60_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for n, x, y, z, i being Variable of g st ex d being Function st ( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds ((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z)))) is_terminating_wrt g
t61_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for n, x, y, z, i being Variable of g st ex d being Function st ( d . b = 0 & d . n = 1 & d . x = 2 & d . y = 3 & d . z = 4 & d . i = 5 ) holds for s being Element of Funcs (X,INT) for N being Element of NAT st N = s . n holds (g . (s,(((x := 0) \; (y := 1)) \; (for-do ((i := 1),(i leq n),(i += 1),(((z := x) \; (x := y)) \; (y += z))))))) . x = Fib N
t62_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y, z being Variable of g st ex d being Function st ( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) is_terminating_wrt g, { s where s is Element of Funcs (X,INT) : ( s . x > s . y & s . y >= 0 ) }
t63_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y, z being Variable of g st ex d being Function st ( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds for s being Element of Funcs (X,INT) for n, m being Element of NAT st n = s . x & m = s . y & n > m holds (g . (s,(while ((y gt 0),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)))))) . x = n gcd m
t64_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y, z being Variable of g st ex d being Function st ( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) is_terminating_wrt g, { s where s is Element of Funcs (X,INT) : ( s . x >= 0 & s . y >= 0 ) }
t65_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y, z being Variable of g st ex d being Function st ( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds for s being Element of Funcs (X,INT) for n, m being Element of NAT st n = s . x & m = s . y & n > 0 holds (g . (s,(while ((y gt 0),((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))))) . x = n gcd m
t66_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y, m being Variable of g st ex d being Function st ( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds (y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x)))) is_terminating_wrt g, { s where s is Element of Funcs (X,INT) : s . m >= 0 }
t67_aofa_i00:: for A being Euclidean preIfWhileAlgebra for X being non empty countable set for b being Element of X for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0) for x, y, m being Variable of g st ex d being Function st ( d . b = 0 & d . x = 1 & d . y = 2 & d . m = 3 ) holds for s being Element of Funcs (X,INT) for n being Nat st n = s . m holds (g . (s,((y := 1) \; (while ((m gt 0),(((if-then ((m is_odd),(y *= x))) \; (m /= 2)) \; (x *= x))))))) . y = (s . x) |^ n
t7_aofa_i00:: for X being set for f being Function holds ( f is Denumeration of X iff ( dom f = card X & rng f = X & f is one-to-one ) )
t8_aofa_i00:: for X being non empty set for x, y being Element of X for f being Enumeration of X holds (f +* (x,(f . y))) +* (y,(f . x)) is Enumeration of X
t9_aofa_i00:: for X being non empty set for x being Element of X ex f being Enumeration of X st f . x = 0
t1_arithm:: for x being complex number holds x + 0 = x
t2_arithm:: for x being complex number holds x * 0 = 0
t3_arithm:: for x being complex number holds 1 * x = x
t4_arithm:: for x being complex number holds x - 0 = x
t5_arithm:: for x being complex number holds 0 / x = 0
t6_arithm:: for x being complex number holds x / 1 = x
d1_armstrng:: for X being set for R being Relation for b3 being Subset of X holds ( b3 = R Maximal_in X iff for x being set holds ( x in b3 iff x is_maximal_wrt X,R ) );
d10_armstrng:: for X being set holds Dependencies-Order X = { [P,Q] where P, Q is Dependency of X : P <= Q } ;
d11_armstrng:: for X being set for F being Dependency-set of X holds ( F is (F1) iff for A being Subset of X holds [A,A] in F );
d12_armstrng:: for X being set for F being Dependency-set of X holds ( F is (F3) iff for A, B, A9, B9 being Subset of X st [A,B] in F & [A,B] >= [A9,B9] holds [A9,B9] in F );
d13_armstrng:: for X being set for F being Dependency-set of X holds ( F is (F4) iff for A, B, A9, B9 being Subset of X st [A,B] in F & [A9,B9] in F holds [(A \/ A9),(B \/ B9)] in F );
d14_armstrng:: for X being set for F being Dependency-set of X holds ( F is full_family iff ( F is (F1) & F is (F2) & F is (F3) & F is (F4) ) );
d15_armstrng:: for X being set for F being Dependency-set of X holds ( F is (DC3) iff for A, B being Subset of X st B c= A holds [A,B] in F );
d16_armstrng:: for X being set for F being Dependency-set of X holds Maximal_wrt F = (Dependencies-Order X) Maximal_in F;
d17_armstrng:: for X being set for F being Dependency-set of X for x, y being set holds ( x ^|^ y,F iff [x,y] in Maximal_wrt F );
d18_armstrng:: for X being set for M being Dependency-set of X holds ( M is (M1) iff for A being Subset of X ex A9, B9 being Subset of X st ( [A9,B9] >= [A,A] & [A9,B9] in M ) );
d19_armstrng:: for X being set for M being Dependency-set of X holds ( M is (M2) iff for A, B, A9, B9 being Subset of X st [A,B] in M & [A9,B9] in M & [A,B] >= [A9,B9] holds ( A = A9 & B = B9 ) );
d2_armstrng:: for x, X being set holds ( x is_/\-irreducible_in X iff ( x in X & ( for z, y being set st z in X & y in X & x = z /\ y & not x = z holds x = y ) ) );
d20_armstrng:: for X being set for M being Dependency-set of X holds ( M is (M3) iff for A, B, A9, B9 being Subset of X st [A,B] in M & [A9,B9] in M & A9 c= B holds B9 c= B );
d21_armstrng:: for X being set for F being Dependency-set of X holds saturated-subsets F = { B where B is Subset of X : ex A being Subset of X st A ^|^ B,F } ;
d22_armstrng:: for X, B being set holds X deps_encl_by B = { [a,b] where a, b is Subset of X : for c being set st c in B & a c= c holds b c= c } ;
d23_armstrng:: for X being set for F being Dependency-set of X holds enclosure_of F = { b where b is Subset of X : for A, B being Subset of X st [A,B] in F & A c= b holds B c= b } ;
d24_armstrng:: for X being non empty finite set for F being Dependency-set of X for b3 being Full-family of X holds ( b3 = Dependency-closure F iff ( F c= b3 & ( for G being Dependency-set of X st F c= G & G is full_family holds b3 c= G ) ) );
d25_armstrng:: for X, G being set for B being Subset-Family of X holds ( G is_generator-set_of B iff ( G c= B & B = { (Intersect S) where S is Subset-Family of X : S c= G } ) );
d26_armstrng:: for X being set for F being Dependency-set of X holds candidate-keys F = { A where A is Subset of X : [A,X] in Maximal_wrt F } ;
d27_armstrng:: for X being set holds ( X is without_proper_subsets iff for x, y being set st x in X & y in X & x c= y holds x = y );
d28_armstrng:: for X being set for F being Dependency-set of X holds ( F is (DC4) iff for A, B, C being Subset of X st [A,B] in F & [A,C] in F holds [A,(B \/ C)] in F );
d29_armstrng:: for X being set for F being Dependency-set of X holds ( F is (DC5) iff for A, B, C, D being Subset of X st [A,B] in F & [(B \/ C),D] in F holds [(A \/ C),D] in F );
d3_armstrng:: for X being non empty set for b2 being Subset of X holds ( b2 = /\-IRR X iff for x being set holds ( x in b2 iff x is_/\-irreducible_in X ) );
d30_armstrng:: for X being set for F being Dependency-set of X holds ( F is (DC6) iff for A, B, C being Subset of X st [A,B] in F holds [(A \/ C),B] in F );
d31_armstrng:: for X being set for F being Dependency-set of X holds charact_set F = { A where A is Subset of X : ex a, b being Subset of X st ( [a,b] in F & a c= A & not b c= A ) } ;
d32_armstrng:: for A, K being set for F being Dependency-set of A holds ( K is_p_i_w_ncv_of F iff ( ( for a being Subset of A st K c= a & a <> A holds a in charact_set F ) & ( for k being set st k c< K holds ex a being Subset of A st ( k c= a & a <> A & not a in charact_set F ) ) ) );
d4_armstrng:: for X being set for B being Subset-Family of X holds ( B is (B1) iff X in B );
d5_armstrng:: for n being Element of NAT for p, q, b4 being Tuple of n, BOOLEAN holds ( b4 = p '&' q iff for i being set st i in Seg n holds b4 . i = (p /. i) '&' (q /. i) );
d6_armstrng:: for X being set holds Dependencies X = [:(bool X),(bool X):];
d7_armstrng:: for R being DB-Rel for A, B being Subset of the Attributes of R holds ( A >|> B,R iff for f, g being Element of the Relationship of R st f | A = g | A holds f | B = g | B );
d8_armstrng:: for R being DB-Rel holds Dependency-str R = { [A,B] where A, B is Subset of the Attributes of R : A >|> B,R } ;
d9_armstrng:: for X being set for P, Q being Dependency of X holds ( P >= Q iff ( P `1 c= Q `1 & Q `2 c= P `2 ) );
s1_armstrng:: scheme SubsetSEq{ F1() -> set , P1[ set ] } : for X1, X2 being Subset of F1() st ( for y being set holds ( y in X1 iff P1[y] ) ) & ( for y being set holds ( y in X2 iff P1[y] ) ) holds X1 = X2
s2_armstrng:: scheme FinIntersect{ F1() -> non empty finite set , P1[ set ] } : for x being set st x in F1() holds P1[x] provided A1: for x being set st x is_/\-irreducible_in F1() holds P1[x] and A2: for x, y being set st x in F1() & y in F1() & P1[x] & P1[y] holds P1[x /\ y]
t1_armstrng:: for B being set st B is cap-closed holds for X being set for S being finite Subset-Family of X st X in B & S c= B holds Intersect S in B
t10_armstrng:: for R being DB-Rel for A, B being Subset of the Attributes of R holds ( [A,B] in Dependency-str R iff A >|> B,R )
t11_armstrng:: for X being set for P, Q being Dependency of X st P <= Q & Q <= P holds P = Q
t12_armstrng:: for X being set for P, Q, S being Dependency of X st P <= Q & Q <= S holds P <= S
t13_armstrng:: for X being set for A, B, A9, B9 being Subset of X holds ( [A,B] >= [A9,B9] iff ( A c= A9 & B9 c= B ) )
t14_armstrng:: for X, x being set holds ( x in Dependencies-Order X iff ex P, Q being Dependency of X st ( x = [P,Q] & P <= Q ) ) ;
t15_armstrng:: for X being set holds dom (Dependencies-Order X) = [:(bool X),(bool X):]
t16_armstrng:: for X being set holds rng (Dependencies-Order X) = [:(bool X),(bool X):]
t17_armstrng:: for X being set holds field (Dependencies-Order X) = [:(bool X),(bool X):]
t18_armstrng:: for X being set for F being Dependency-set of X holds ( F is (F2) iff for A, B, C being Subset of X st [A,B] in F & [B,C] in F holds [A,C] in F )
t19_armstrng:: for X being set holds ( Dependencies X is (F1) & Dependencies X is (F2) & Dependencies X is (F3) & Dependencies X is (F4) )
t2_armstrng:: for R being non empty antisymmetric transitive Relation for X being finite Subset of (field R) st X <> {} holds ex x being Element of X st x is_maximal_wrt X,R
t20_armstrng:: for X being finite set for F being Dependency-set of X holds F is finite ;
t21_armstrng:: for X being set for F being Dependency-set of X st F is (DC3) & F is (F2) holds ( F is (F1) & F is (F3) ) ;
t22_armstrng:: for X being set for F being Dependency-set of X st F is (F1) & F is (F3) holds F is (DC3) ;
t23_armstrng:: for R being DB-Rel holds Dependency-str R is full_family
t24_armstrng:: for X being set for K being Subset of X holds { [A,B] where A, B is Subset of X : ( K c= A or B c= A ) } is Full-family of X
t25_armstrng:: for X being set for F being Dependency-set of X holds Maximal_wrt F c= F ;
t26_armstrng:: for X being finite set for P being Dependency of X for F being Dependency-set of X st P in F holds ex A, B being Subset of X st ( [A,B] in Maximal_wrt F & [A,B] >= P )
t27_armstrng:: for X being set for F being Dependency-set of X for A, B being Subset of X holds ( A ^|^ B,F iff ( [A,B] in F & ( for A9, B9 being Subset of X holds ( not [A9,B9] in F or ( not A <> A9 & not B <> B9 ) or not [A,B] <= [A9,B9] ) ) ) )
t28_armstrng:: for X being non empty finite set for F being Full-family of X holds ( Maximal_wrt F is (M1) & Maximal_wrt F is (M2) & Maximal_wrt F is (M3) )
t29_armstrng:: for X being finite set for M, F being Dependency-set of X st M is (M1) & M is (M2) & M is (M3) & F = { [A,B] where A, B is Subset of X : ex A9, B9 being Subset of X st ( [A9,B9] >= [A,B] & [A9,B9] in M ) } holds ( M = Maximal_wrt F & F is full_family & ( for G being Full-family of X st M = Maximal_wrt G holds G = F ) )
t3_armstrng:: for X being non empty finite set for x being Element of X ex A being non empty Subset of X st ( x = meet A & ( for s being set st s in A holds s is_/\-irreducible_in X ) )
t30_armstrng:: for X being finite set for F being Dependency-set of X for K being Subset of X st F = { [A,B] where A, B is Subset of X : ( K c= A or B c= A ) } holds {[K,X]} \/ { [A,A] where A is Subset of X : not K c= A } = Maximal_wrt F
t31_armstrng:: for X, x being set for F being Dependency-set of X holds ( x in saturated-subsets F iff ex B, A being Subset of X st ( x = B & A ^|^ B,F ) )
t32_armstrng:: for X being non empty finite set for F being Full-family of X holds ( saturated-subsets F is (B1) & saturated-subsets F is (B2) )
t33_armstrng:: for X being set for B being Subset-Family of X for F being Dependency-set of X holds X deps_encl_by B is full_family
t34_armstrng:: for X being non empty finite set for B being Subset-Family of X holds B c= saturated-subsets (X deps_encl_by B)
t35_armstrng:: for X being non empty finite set for B being Subset-Family of X st B is (B1) & B is (B2) holds ( B = saturated-subsets (X deps_encl_by B) & ( for G being Full-family of X st B = saturated-subsets G holds G = X deps_encl_by B ) )
t36_armstrng:: for X being non empty finite set for F being Dependency-set of X holds ( enclosure_of F is (B1) & enclosure_of F is (B2) )
t37_armstrng:: for X being non empty finite set for F being Dependency-set of X holds ( F c= X deps_encl_by (enclosure_of F) & ( for G being Dependency-set of X st F c= G & G is full_family holds X deps_encl_by (enclosure_of F) c= G ) )
t38_armstrng:: for X being non empty finite set for F being Dependency-set of X holds Dependency-closure F = X deps_encl_by (enclosure_of F)
t39_armstrng:: for X being set for K being Subset of X for B being Subset-Family of X st B = {X} \/ { A where A is Subset of X : not K c= A } holds ( B is (B1) & B is (B2) )
t4_armstrng:: for X being set for B being non empty Subset-Family of X st B is cap-closed holds for x being Element of B st x is_/\-irreducible_in B & x <> X holds for S being finite Subset-Family of X st S c= B & x = Intersect S holds x in S
t40_armstrng:: for X being non empty finite set for F being Dependency-set of X for K being Subset of X st F = { [A,B] where A, B is Subset of X : ( K c= A or B c= A ) } holds {X} \/ { B where B is Subset of X : not K c= B } = saturated-subsets F
t41_armstrng:: for X being finite set for F being Dependency-set of X for K being Subset of X st F = { [A,B] where A, B is Subset of X : ( K c= A or B c= A ) } holds {X} \/ { B where B is Subset of X : not K c= B } = saturated-subsets F
t42_armstrng:: for X being non empty finite set for G being Subset-Family of X holds G is_generator-set_of saturated-subsets (X deps_encl_by G)
t43_armstrng:: for X being non empty finite set for F being Full-family of X ex G being Subset-Family of X st ( G is_generator-set_of saturated-subsets F & F = X deps_encl_by G )
t44_armstrng:: for X being set for B being non empty finite Subset-Family of X st B is (B1) & B is (B2) holds /\-IRR B is_generator-set_of B
t45_armstrng:: for X, G being set for B being non empty finite Subset-Family of X st B is (B1) & B is (B2) & G is_generator-set_of B holds /\-IRR B c= G \/ {X}
t46_armstrng:: for X being non empty finite set for F being Full-family of X ex R being DB-Rel st ( the Attributes of R = X & ( for a being Element of X holds the Domains of R . a = INT ) & F = Dependency-str R )
t47_armstrng:: for X being finite set for F being Dependency-set of X for K being Subset of X st F = { [A,B] where A, B is Subset of X : ( K c= A or B c= A ) } holds candidate-keys F = {K}
t48_armstrng:: for R being DB-Rel holds ( candidate-keys (Dependency-str R) is (C1) & candidate-keys (Dependency-str R) is (C2) )
t49_armstrng:: for X being finite set for C being Subset-Family of X st C is (C1) & C is (C2) holds ex F being Full-family of X st C = candidate-keys F
t5_armstrng:: for n being Element of NAT for p being Element of n -tuples_on BOOLEAN holds (n -BinarySequence 0) '&' p = n -BinarySequence 0
t50_armstrng:: for X being finite set for C being Subset-Family of X for B being set st C is (C1) & C is (C2) & B = { b where b is Subset of X : for K being Subset of X st K in C holds not K c= b } holds C = candidate-keys (X deps_encl_by B)
t51_armstrng:: for X being non empty finite set for C being Subset-Family of X st C is (C1) & C is (C2) holds ex R being DB-Rel st ( the Attributes of R = X & C = candidate-keys (Dependency-str R) )
t52_armstrng:: for X being set for F being Dependency-set of X holds ( F is (F1) & F is (F2) & F is (F3) & F is (F4) iff ( F is (F2) & F is (DC3) & F is (F4) ) ) ;
t53_armstrng:: for X being set for F being Dependency-set of X holds ( F is (F1) & F is (F2) & F is (F3) & F is (F4) iff ( F is (DC1) & F is (DC3) & F is (DC4) ) )
t54_armstrng:: for X being set for F being Dependency-set of X holds ( F is (F1) & F is (F2) & F is (F3) & F is (F4) iff ( F is (DC2) & F is (DC5) & F is (DC6) ) )
t55_armstrng:: for X, A being set for F being Dependency-set of X st A in charact_set F holds ( A is Subset of X & ex a, b being Subset of X st ( [a,b] in F & a c= A & not b c= A ) )
t56_armstrng:: for X being set for A being Subset of X for F being Dependency-set of X st ex a, b being Subset of X st ( [a,b] in F & a c= A & not b c= A ) holds A in charact_set F ;
t57_armstrng:: for X being non empty finite set for F being Dependency-set of X holds ( ( for A, B being Subset of X holds ( [A,B] in Dependency-closure F iff for a being Subset of X st A c= a & not B c= a holds a in charact_set F ) ) & saturated-subsets (Dependency-closure F) = (bool X) \ (charact_set F) )
t58_armstrng:: for X being non empty finite set for F, G being Dependency-set of X st charact_set F = charact_set G holds Dependency-closure F = Dependency-closure G
t59_armstrng:: for X being non empty finite set for F being Dependency-set of X holds charact_set F = charact_set (Dependency-closure F)
t6_armstrng:: for n being Element of NAT for p being Tuple of n, BOOLEAN holds ('not' (n -BinarySequence 0)) '&' p = p
t60_armstrng:: for X being non empty finite set for F being Dependency-set of X for K being Subset of X holds ( K in candidate-keys (Dependency-closure F) iff K is_p_i_w_ncv_of F )
t7_armstrng:: for n, i being Element of NAT st i < n holds ( (n -BinarySequence (2 to_power i)) . (i + 1) = 1 & ( for j being Element of NAT st j in Seg n & j <> i + 1 holds (n -BinarySequence (2 to_power i)) . j = 0 ) )
t8_armstrng:: for X, x being set holds ( x in Dependencies X iff ex a, b being Subset of X st x = [a,b] )
t9_armstrng:: for X, x being set for F being Dependency-set of X st x in F holds ex a, b being Subset of X st x = [a,b]
d1_arrow:: for A being non empty set for b2 being set holds ( b2 = LinPreorders A iff for R being set holds ( R in b2 iff ( R is Relation of A & ( for a, b being Element of A holds ( [a,b] in R or [b,a] in R ) ) & ( for a, b, c being Element of A st [a,b] in R & [b,c] in R holds [a,c] in R ) ) ) );
d2_arrow:: for A being non empty set for b2 being Subset of (LinPreorders A) holds ( b2 = LinOrders A iff for R being Element of LinPreorders A holds ( R in b2 iff for a, b being Element of A st [a,b] in R & [b,a] in R holds a = b ) );
d3_arrow:: for A being non empty set for b2 being Subset of (LinPreorders A) holds ( b2 = LinOrders A iff for R being set holds ( R in b2 iff R is connected Order of A ) );
d4_arrow:: for o being Relation for a, b being set holds ( a <=_ o,b iff [a,b] in o );
t1_arrow:: for A being finite set st card A >= 2 holds for a being Element of A ex b being Element of A st b <> a
t10_arrow:: for A being non empty set for a, b, c being Element of A for o9 being Element of LinPreorders A st a <> b & a <> c holds ex o being Element of LinPreorders A st ( a <_ o,b & a <_ o,c & ( b <_ o,c implies b <_ o9,c ) & ( b <_ o9,c implies b <_ o,c ) & ( c <_ o,b implies c <_ o9,b ) & ( c <_ o9,b implies c <_ o,b ) )
t11_arrow:: for A being non empty set for a, b, c being Element of A for o9 being Element of LinPreorders A st a <> b & a <> c holds ex o being Element of LinPreorders A st ( b <_ o,a & c <_ o,a & ( b <_ o,c implies b <_ o9,c ) & ( b <_ o9,c implies b <_ o,c ) & ( c <_ o,b implies c <_ o9,b ) & ( c <_ o9,b implies c <_ o,b ) )
t12_arrow:: for A being non empty set for a, b being Element of A for o, o9 being Element of LinOrders A holds ( not ( ( a <_ o,b implies a <_ o9,b ) & ( a <_ o9,b implies a <_ o,b ) & ( b <_ o,a implies b <_ o9,a ) & ( b <_ o9,a implies b <_ o,a ) & not ( a <_ o,b iff a <_ o9,b ) ) & not ( ( a <_ o,b implies a <_ o9,b ) & ( a <_ o9,b implies a <_ o,b ) & not ( ( a <_ o,b implies a <_ o9,b ) & ( a <_ o9,b implies a <_ o,b ) & ( b <_ o,a implies b <_ o9,a ) & ( b <_ o9,a implies b <_ o,a ) ) ) )
t13_arrow:: for A being non empty set for o being Element of LinOrders A for o9 being Element of LinPreorders A holds ( ( for a, b being Element of A st a <_ o,b holds a <_ o9,b ) iff for a, b being Element of A holds ( a <_ o,b iff a <_ o9,b ) )
t14_arrow:: for A, N being non empty finite set for f being Function of (Funcs (N,(LinPreorders A))),(LinPreorders A) st ( for p being Element of Funcs (N,(LinPreorders A)) for a, b being Element of A st ( for i being Element of N holds a <_ p . i,b ) holds a <_ f . p,b ) & ( for p, p9 being Element of Funcs (N,(LinPreorders A)) for a, b being Element of A st ( for i being Element of N holds ( ( a <_ p . i,b implies a <_ p9 . i,b ) & ( a <_ p9 . i,b implies a <_ p . i,b ) & ( b <_ p . i,a implies b <_ p9 . i,a ) & ( b <_ p9 . i,a implies b <_ p . i,a ) ) ) holds ( a <_ f . p,b iff a <_ f . p9,b ) ) & card A >= 3 holds ex n being Element of N st for p being Element of Funcs (N,(LinPreorders A)) for a, b being Element of A st a <_ p . n,b holds a <_ f . p,b
t15_arrow:: for A, N being non empty finite set for f being Function of (Funcs (N,(LinOrders A))),(LinPreorders A) st ( for p being Element of Funcs (N,(LinOrders A)) for a, b being Element of A st ( for i being Element of N holds a <_ p . i,b ) holds a <_ f . p,b ) & ( for p, p9 being Element of Funcs (N,(LinOrders A)) for a, b being Element of A st ( for i being Element of N holds ( a <_ p . i,b iff a <_ p9 . i,b ) ) holds ( a <_ f . p,b iff a <_ f . p9,b ) ) & card A >= 3 holds ex n being Element of N st for p being Element of Funcs (N,(LinOrders A)) for a, b being Element of A holds ( a <_ p . n,b iff a <_ f . p,b )
t2_arrow:: for A being finite set st card A >= 3 holds for a, b being Element of A ex c being Element of A st ( c <> a & c <> b )
t3_arrow:: for A being non empty set for a being Element of A for o being Element of LinPreorders A holds a <=_ o,a
t4_arrow:: for A being non empty set for a, b being Element of A for o being Element of LinPreorders A holds ( a <=_ o,b or b <=_ o,a )
t5_arrow:: for A being non empty set for a, b, c being Element of A for o being Element of LinPreorders A st ( a <=_ o,b or a <_ o,b ) & ( b <=_ o,c or b <_ o,c ) holds a <=_ o,c
t6_arrow:: for A being non empty set for a, b being Element of A for o99 being Element of LinOrders A st a <=_ o99,b & b <=_ o99,a holds a = b
t7_arrow:: for A being non empty set for a, b, c being Element of A st a <> b & b <> c & a <> c holds ex o being Element of LinPreorders A st ( a <_ o,b & b <_ o,c )
t8_arrow:: for A being non empty set for b being Element of A ex o being Element of LinPreorders A st for a being Element of A st a <> b holds b <_ o,a
t9_arrow:: for A being non empty set for b being Element of A ex o being Element of LinPreorders A st for a being Element of A st a <> b holds a <_ o,b
d1_arytm_0:: for x, y, b3 being Element of REAL holds ( ( x in REAL+ & y in REAL+ implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st ( x = x9 & y = y9 & b3 = x9 + y9 ) ) ) & ( x in REAL+ & y in [:{0},REAL+:] implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st ( x = x9 & y = [0,y9] & b3 = x9 - y9 ) ) ) & ( y in REAL+ & x in [:{0},REAL+:] implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st ( x = [0,x9] & y = y9 & b3 = y9 - x9 ) ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in [:{0},REAL+:] ) & ( not y in REAL+ or not x in [:{0},REAL+:] ) implies ( b3 = + (x,y) iff ex x9, y9 being Element of REAL+ st ( x = [0,x9] & y = [0,y9] & b3 = [0,(x9 + y9)] ) ) ) );
d2_arytm_0:: for x, y, b3 being Element of REAL holds ( ( x in REAL+ & y in REAL+ implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st ( x = x9 & y = y9 & b3 = x9 *' y9 ) ) ) & ( x in REAL+ & y in [:{0},REAL+:] & x <> 0 implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st ( x = x9 & y = [0,y9] & b3 = [0,(x9 *' y9)] ) ) ) & ( y in REAL+ & x in [:{0},REAL+:] & y <> 0 implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st ( x = [0,x9] & y = y9 & b3 = [0,(y9 *' x9)] ) ) ) & ( x in [:{0},REAL+:] & y in [:{0},REAL+:] implies ( b3 = * (x,y) iff ex x9, y9 being Element of REAL+ st ( x = [0,x9] & y = [0,y9] & b3 = y9 *' x9 ) ) ) & ( ( not x in REAL+ or not y in REAL+ ) & ( not x in REAL+ or not y in [:{0},REAL+:] or not x <> 0 ) & ( not y in REAL+ or not x in [:{0},REAL+:] or not y <> 0 ) & ( not x in [:{0},REAL+:] or not y in [:{0},REAL+:] ) implies ( b3 = * (x,y) iff b3 = 0 ) ) );
d3_arytm_0:: for x, b2 being Element of REAL holds ( b2 = opp x iff + (x,b2) = 0 );
d4_arytm_0:: for x, b2 being Element of REAL holds ( ( x <> 0 implies ( b2 = inv x iff * (x,b2) = 1 ) ) & ( not x <> 0 implies ( b2 = inv x iff b2 = 0 ) ) );
d5_arytm_0:: for x, y being Element of REAL holds ( ( y = 0 implies [*x,y*] = x ) & ( not y = 0 implies [*x,y*] = (0,1) --> (x,y) ) );
t1_arytm_0:: REAL+ c= REAL
t10_arytm_0:: for x1, x2, y1, y2 being Element of REAL st [*x1,x2*] = [*y1,y2*] holds ( x1 = y1 & x2 = y2 )
t11_arytm_0:: for x, o being Element of REAL st o = 0 holds + (x,o) = x
t12_arytm_0:: for x, o being Element of REAL st o = 0 holds * (x,o) = 0
t13_arytm_0:: for x, y, z being Element of REAL holds * (x,(* (y,z))) = * ((* (x,y)),z)
t14_arytm_0:: for x, y, z being Element of REAL holds * (x,(+ (y,z))) = + ((* (x,y)),(* (x,z)))
t15_arytm_0:: for x, y being Element of REAL holds * ((opp x),y) = opp (* (x,y))
t16_arytm_0:: for x being Element of REAL holds * (x,x) in REAL+
t17_arytm_0:: for x, y being Element of REAL st + ((* (x,x)),(* (y,y))) = 0 holds x = 0
t18_arytm_0:: for x, y, z being Element of REAL st x <> 0 & * (x,y) = 1 & * (x,z) = 1 holds y = z
t19_arytm_0:: for x, y being Element of REAL st y = 1 holds * (x,y) = x
t2_arytm_0:: for x being Element of REAL+ st x <> {} holds [{},x] in REAL
t20_arytm_0:: for x, y being Element of REAL st y <> 0 holds * ((* (x,y)),(inv y)) = x
t21_arytm_0:: for x, y being Element of REAL holds ( not * (x,y) = 0 or x = 0 or y = 0 )
t22_arytm_0:: for x, y being Element of REAL holds inv (* (x,y)) = * ((inv x),(inv y))
t23_arytm_0:: for x, y, z being Element of REAL holds + (x,(+ (y,z))) = + ((+ (x,y)),z)
t24_arytm_0:: for x, y being Element of REAL st [*x,y*] in REAL holds y = 0
t25_arytm_0:: for x, y being Element of REAL holds opp (+ (x,y)) = + ((opp x),(opp y))
t3_arytm_0:: for y being set st [{},y] in REAL holds y <> {}
t4_arytm_0:: for x, y being Element of REAL+ holds x - y in REAL
t5_arytm_0:: REAL+ misses [:{{}},REAL+:]
t6_arytm_0:: for x, y being Element of REAL+ st x - y = {} holds x = y
t7_arytm_0:: for x, y, z being Element of REAL+ st x <> {} & x *' y = x *' z holds y = z
t8_arytm_0:: for a, b being Element of REAL holds not (0,1) --> (a,b) in REAL
t9_arytm_0:: for c being Element of COMPLEX ex r, s being Element of REAL st c = [*r,s*]
d1_arytm_1:: for x, y, b3 being Element of REAL+ holds ( ( y <=' x implies ( b3 = x -' y iff b3 + y = x ) ) & ( not y <=' x implies ( b3 = x -' y iff b3 = {} ) ) );
d2_arytm_1:: for x, y being Element of REAL+ holds ( ( y <=' x implies x - y = x -' y ) & ( not y <=' x implies x - y = [{},(y -' x)] ) );
t1_arytm_1:: for x, y being Element of REAL+ st x + y = y holds x = {}
t10_arytm_1:: for x, y being Element of REAL+ st x <=' y & y -' x = {} holds x = y
t11_arytm_1:: for x, y being Element of REAL+ holds x -' y <=' x
t12_arytm_1:: for y, x, z being Element of REAL+ st y <=' x & y <=' z holds x + (z -' y) = (x -' y) + z
t13_arytm_1:: for z, y, x being Element of REAL+ st z <=' y holds x + (y -' z) = (x + y) -' z
t14_arytm_1:: for z, x, y being Element of REAL+ st z <=' x & y <=' z holds (x -' z) + y = x -' (z -' y)
t15_arytm_1:: for y, x, z being Element of REAL+ st y <=' x & y <=' z holds (z -' y) + x = (x -' y) + z
t16_arytm_1:: for x, y, z being Element of REAL+ st x <=' y holds z -' y <=' z -' x
t17_arytm_1:: for x, y, z being Element of REAL+ st x <=' y holds x -' z <=' y -' z
t18_arytm_1:: for x being Element of REAL+ holds x - x = {}
t19_arytm_1:: for x, y being Element of REAL+ st x = {} & y <> {} holds x - y = [{},y]
t2_arytm_1:: for x, y being Element of REAL+ holds ( not x *' y = {} or x = {} or y = {} )
t20_arytm_1:: for z, y, x being Element of REAL+ st z <=' y holds x + (y -' z) = (x + y) - z
t21_arytm_1:: for z, y, x being Element of REAL+ st not z <=' y holds x - (z -' y) = (x + y) - z
t22_arytm_1:: for y, x, z being Element of REAL+ st y <=' x & not y <=' z holds x - (y -' z) = (x -' y) + z
t23_arytm_1:: for y, x, z being Element of REAL+ st not y <=' x & not y <=' z holds x - (y -' z) = z - (y -' x)
t24_arytm_1:: for y, x, z being Element of REAL+ st y <=' x holds x - (y + z) = (x -' y) - z
t25_arytm_1:: for x, y, z being Element of REAL+ st x <=' y & z <=' y holds (y -' z) - x = (y -' x) - z
t26_arytm_1:: for z, y, x being Element of REAL+ st z <=' y holds x *' (y -' z) = (x *' y) - (x *' z)
t27_arytm_1:: for z, y, x being Element of REAL+ st not z <=' y & x <> {} holds [{},(x *' (z -' y))] = (x *' y) - (x *' z)
t28_arytm_1:: for y, z, x being Element of REAL+ st y -' z <> {} & z <=' y & x <> {} holds (x *' z) - (x *' y) = [{},(x *' (y -' z))]
t3_arytm_1:: for x, y, z being Element of REAL+ st x <=' y & y <=' z holds x <=' z
t4_arytm_1:: for x, y being Element of REAL+ st x <=' y & y <=' x holds x = y
t5_arytm_1:: for x, y being Element of REAL+ st x <=' y & y = {} holds x = {}
t6_arytm_1:: for x, y being Element of REAL+ st x = {} holds x <=' y
t7_arytm_1:: for x, y, z being Element of REAL+ holds ( x <=' y iff x + z <=' y + z )
t8_arytm_1:: for x, y, z being Element of REAL+ st x <=' y holds x *' z <=' y *' z
t9_arytm_1:: for x, y being Element of REAL+ holds ( x <=' y or x -' y <> {} )
d1_arytm_2:: DEDEKIND_CUTS = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) } \ {RAT+};
d2_arytm_2:: REAL+ = (RAT+ \/ DEDEKIND_CUTS) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;
d3_arytm_2:: for x being Element of REAL+ for b2 being Element of DEDEKIND_CUTS holds ( ( x in RAT+ implies ( b2 = DEDEKIND_CUT x iff ex r being Element of RAT+ st ( x = r & b2 = { s where s is Element of RAT+ : s < r } ) ) ) & ( not x in RAT+ implies ( b2 = DEDEKIND_CUT x iff b2 = x ) ) );
d4_arytm_2:: for x being Element of DEDEKIND_CUTS for b2 being Element of REAL+ holds ( ( ex r being Element of RAT+ st for s being Element of RAT+ holds ( s in x iff s < r ) implies ( b2 = GLUED x iff ex r being Element of RAT+ st ( b2 = r & ( for s being Element of RAT+ holds ( s in x iff s < r ) ) ) ) ) & ( ( for r being Element of RAT+ holds not for s being Element of RAT+ holds ( s in x iff s < r ) ) implies ( b2 = GLUED x iff b2 = x ) ) );
d5_arytm_2:: for x, y being Element of REAL+ holds ( ( x in RAT+ & y in RAT+ implies ( x <=' y iff ex x9, y9 being Element of RAT+ st ( x = x9 & y = y9 & x9 <=' y9 ) ) ) & ( x in RAT+ & not y in RAT+ implies ( x <=' y iff x in y ) ) & ( not x in RAT+ & y in RAT+ implies ( x <=' y iff not y in x ) ) & ( ( not x in RAT+ or not y in RAT+ ) & ( not x in RAT+ or y in RAT+ ) & ( x in RAT+ or not y in RAT+ ) implies ( x <=' y iff x c= y ) ) );
d6_arytm_2:: for A, B being Element of DEDEKIND_CUTS holds A + B = { (r + s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;
d7_arytm_2:: for A, B being Element of DEDEKIND_CUTS holds A *' B = { (r *' s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;
d8_arytm_2:: for x, y being Element of REAL+ holds ( ( y = {} implies x + y = x ) & ( x = {} implies x + y = y ) & ( not y = {} & not x = {} implies x + y = GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y)) ) );
d9_arytm_2:: for x, y being Element of REAL+ holds x *' y = GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y));
t1_arytm_2:: RAT+ c= REAL+ by Lm2, XBOOLE_1:7, XBOOLE_1:86;
t10_arytm_2:: for x, y being Element of REAL+ ex z being Element of REAL+ st ( x + z = y or y + z = x )
t11_arytm_2:: for x, y, z being Element of REAL+ st x + y = x + z holds y = z
t12_arytm_2:: for x, y, z being Element of REAL+ holds x *' (y *' z) = (x *' y) *' z
t13_arytm_2:: for x, y, z being Element of REAL+ holds x *' (y + z) = (x *' y) + (x *' z)
t14_arytm_2:: for x being Element of REAL+ st x <> {} holds ex y being Element of REAL+ st x *' y = one
t15_arytm_2:: for x, y being Element of REAL+ st x = one holds x *' y = y
t16_arytm_2:: for x, y being Element of REAL+ st x in omega & y in omega holds y + x in omega
t17_arytm_2:: for A being Subset of REAL+ st {} in A & ( for x, y being Element of REAL+ st x in A & y = one holds x + y in A ) holds omega c= A
t18_arytm_2:: for x being Element of REAL+ st x in omega holds for y being Element of REAL+ holds ( y in x iff ( y in omega & y <> x & y <=' x ) )
t19_arytm_2:: for x, y, z being Element of REAL+ st x = y + z holds z <=' x
t2_arytm_2:: omega c= REAL+ by Lm5, Th1, XBOOLE_1:1;
t20_arytm_2:: ( {} in REAL+ & one in REAL+ )
t21_arytm_2:: for x, y being Element of REAL+ st x in RAT+ & y in RAT+ holds ex x9, y9 being Element of RAT+ st ( x = x9 & y = y9 & x *' y = x9 *' y9 )
t3_arytm_2:: for y being set holds not [{},y] in REAL+
t4_arytm_2:: for x, y being Element of REAL+ st x = {} holds x *' y = {}
t5_arytm_2:: for x, y being Element of REAL+ st x + y = {} holds x = {}
t6_arytm_2:: for x, y, z being Element of REAL+ holds x + (y + z) = (x + y) + z
t7_arytm_2:: { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds ( ( for s being Element of RAT+ st s <=' r holds s in A ) & ex s being Element of RAT+ st ( s in A & r < s ) ) } is c=-linear
t8_arytm_2:: for X, Y being Subset of REAL+ st ex x being Element of REAL+ st x in Y & ( for x, y being Element of REAL+ st x in X & y in Y holds x <=' y ) holds ex z being Element of REAL+ st for x, y being Element of REAL+ st x in X & y in Y holds ( x <=' z & z <=' y )
t9_arytm_2:: for x, y being Element of REAL+ st x <=' y holds ex z being Element of REAL+ st x + z = y
d1_arytm_3:: one = 1;
d10_arytm_3:: for i, j being natural Ordinal holds ( ( j = {} implies i / j = {} ) & ( RED (j,i) = 1 implies i / j = RED (i,j) ) & ( not j = {} & not RED (j,i) = 1 implies i / j = [(RED (i,j)),(RED (j,i))] ) );
d11_arytm_3:: for x, y being Element of RAT+ holds x + y = (((numerator x) *^ (denominator y)) +^ ((numerator y) *^ (denominator x))) / ((denominator x) *^ (denominator y));
d12_arytm_3:: for x, y being Element of RAT+ holds x *' y = ((numerator x) *^ (numerator y)) / ((denominator x) *^ (denominator y));
d13_arytm_3:: for x, y being Element of RAT+ holds ( x <=' y iff ex z being Element of RAT+ st y = x + z );
d2_arytm_3:: for a, b being Ordinal holds ( a,b are_relative_prime iff for c, d1, d2 being Ordinal st a = c *^ d1 & b = c *^ d2 holds c = 1 );
d3_arytm_3:: for k, n being Ordinal holds ( k divides n iff ex a being Ordinal st n = k *^ a );
d4_arytm_3:: for k, n being natural Ordinal for b3 being Element of omega holds ( b3 = k lcm n iff ( k divides b3 & n divides b3 & ( for m being natural Ordinal st k divides m & n divides m holds b3 divides m ) ) );
d5_arytm_3:: for k, n being natural Ordinal for b3 being Element of omega holds ( b3 = k hcf n iff ( b3 divides k & b3 divides n & ( for m being natural Ordinal st m divides k & m divides n holds m divides b3 ) ) );
d6_arytm_3:: for a, b being natural Ordinal holds RED (a,b) = a div^ (a hcf b);
d7_arytm_3:: RAT+ = ( { [i,j] where i, j is Element of omega : ( i,j are_relative_prime & j <> {} ) } \ { [k,1] where k is Element of omega : verum } ) \/ omega;
d8_arytm_3:: for x being Element of RAT+ for b2 being Element of omega holds ( ( x in omega implies ( b2 = numerator x iff b2 = x ) ) & ( not x in omega implies ( b2 = numerator x iff ex a being natural Ordinal st x = [b2,a] ) ) );
d9_arytm_3:: for x being Element of RAT+ for b2 being Element of omega holds ( ( x in omega implies ( b2 = denominator x iff b2 = 1 ) ) & ( not x in omega implies ( b2 = denominator x iff ex a being natural Ordinal st x = [a,b2] ) ) );
s1_arytm_3:: scheme DisNat{ F1() -> Element of RAT+ , F2() -> Element of RAT+ , F3() -> Element of RAT+ , P1[ set ] } : ex s being Element of RAT+ st ( s in omega & P1[s] & P1[s + F2()] ) provided A1: F2() = 1 and A2: F1() = {} and A3: F3() in omega and A4: P1[F1()] and A5: P1[F3()]
t1_arytm_3:: not {} , {} are_relative_prime
t10_arytm_3:: for n, m being natural Ordinal st {} in m & n divides m holds n c= m
t11_arytm_3:: for n, m, l being natural Ordinal st n divides m & n divides m +^ l holds n divides l
t12_arytm_3:: for m, n being natural Ordinal holds m lcm n divides m *^ n
t13_arytm_3:: for n, m being natural Ordinal st n <> {} holds (m *^ n) div^ (m lcm n) divides m
t14_arytm_3:: for a being natural Ordinal holds ( a hcf {} = a & a lcm {} = {} )
t15_arytm_3:: for a, b being natural Ordinal st a hcf b = {} holds a = {}
t16_arytm_3:: for a being natural Ordinal holds ( a hcf a = a & a lcm a = a )
t17_arytm_3:: for a, c, b being natural Ordinal holds (a *^ c) hcf (b *^ c) = (a hcf b) *^ c
t18_arytm_3:: for b, a being natural Ordinal st b <> {} holds ( a hcf b <> {} & b div^ (a hcf b) <> {} )
t19_arytm_3:: for a, b being natural Ordinal st ( a <> {} or b <> {} ) holds a div^ (a hcf b),b div^ (a hcf b) are_relative_prime
t2_arytm_3:: for A being Ordinal holds 1,A are_relative_prime
t20_arytm_3:: for a, b being natural Ordinal holds ( a,b are_relative_prime iff a hcf b = 1 )
t21_arytm_3:: for a, b being natural Ordinal holds (RED (a,b)) *^ (a hcf b) = a
t22_arytm_3:: for a, b being natural Ordinal st ( a <> {} or b <> {} ) holds RED (a,b), RED (b,a) are_relative_prime by Th19;
t23_arytm_3:: for a, b being natural Ordinal st a,b are_relative_prime holds RED (a,b) = a
t24_arytm_3:: for a being natural Ordinal holds ( RED (a,1) = a & RED (1,a) = 1 )
t25_arytm_3:: for b, a being natural Ordinal st b <> {} holds RED (b,a) <> {} by Th18;
t26_arytm_3:: for a being natural Ordinal holds ( RED ({},a) = {} & ( a <> {} implies RED (a,{}) = 1 ) )
t27_arytm_3:: for a being natural Ordinal st a <> {} holds RED (a,a) = 1
t28_arytm_3:: for c, a, b being natural Ordinal st c <> {} holds RED ((a *^ c),(b *^ c)) = RED (a,b)
t29_arytm_3:: for x being Element of RAT+ holds ( x in omega or ex i, j being Element of omega st ( x = [i,j] & i,j are_relative_prime & j <> {} & j <> 1 ) )
t3_arytm_3:: for A being Ordinal st {} ,A are_relative_prime holds A = 1
t30_arytm_3:: for i, j being set holds [i,j] is not Ordinal
t31_arytm_3:: for A being Ordinal st A in RAT+ holds A in omega
t32_arytm_3:: for i, j being set holds not [i,j] in omega
t33_arytm_3:: for i, j being Element of omega holds ( [i,j] in RAT+ iff ( i,j are_relative_prime & j <> {} & j <> 1 ) )
t34_arytm_3:: for x being Element of RAT+ holds numerator x, denominator x are_relative_prime
t35_arytm_3:: for x being Element of RAT+ holds denominator x <> {}
t36_arytm_3:: for x being Element of RAT+ st not x in omega holds ( x = [(numerator x),(denominator x)] & denominator x <> 1 )
t37_arytm_3:: for x being Element of RAT+ holds ( x <> {} iff numerator x <> {} )
t38_arytm_3:: for x being Element of RAT+ holds ( x in omega iff denominator x = 1 ) by Def9, Th36;
t39_arytm_3:: for x being Element of RAT+ holds (numerator x) / (denominator x) = x
t4_arytm_3:: for a, b being natural Ordinal st ( a <> {} or b <> {} ) holds ex c, d1, d2 being natural Ordinal st ( d1,d2 are_relative_prime & a = c *^ d1 & b = c *^ d2 )
t40_arytm_3:: for b, a being natural Ordinal holds ( {} / b = {} & a / 1 = a )
t41_arytm_3:: for a being natural Ordinal st a <> {} holds a / a = 1
t42_arytm_3:: for b, a being natural Ordinal st b <> {} holds ( numerator (a / b) = RED (a,b) & denominator (a / b) = RED (b,a) )
t43_arytm_3:: for i, j being Element of omega st i,j are_relative_prime & j <> {} holds ( numerator (i / j) = i & denominator (i / j) = j )
t44_arytm_3:: for c, a, b being natural Ordinal st c <> {} holds (a *^ c) / (b *^ c) = a / b
t45_arytm_3:: for l, j, i, k being natural Ordinal st j <> {} & l <> {} holds ( i / j = k / l iff i *^ l = j *^ k )
t46_arytm_3:: for l, j, i, k being natural Ordinal st j <> {} & l <> {} holds (i / j) + (k / l) = ((i *^ l) +^ (j *^ k)) / (j *^ l)
t47_arytm_3:: for k, i, j being natural Ordinal st k <> {} holds (i / k) + (j / k) = (i +^ j) / k
t48_arytm_3:: for x being Element of RAT+ holds x *' {} = {}
t49_arytm_3:: for l, i, j, k being natural Ordinal holds (i / j) *' (k / l) = (i *^ k) / (j *^ l)
t5_arytm_3:: for a, b being natural Ordinal holds ( a divides b iff ex c being natural Ordinal st b = a *^ c )
t50_arytm_3:: for x being Element of RAT+ holds x + {} = x
t51_arytm_3:: for x, y, z being Element of RAT+ holds (x + y) + z = x + (y + z)
t52_arytm_3:: for x, y, z being Element of RAT+ holds (x *' y) *' z = x *' (y *' z)
t53_arytm_3:: for x being Element of RAT+ holds x *' one = x
t54_arytm_3:: for x being Element of RAT+ st x <> {} holds ex y being Element of RAT+ st x *' y = 1
t55_arytm_3:: for x, y being Element of RAT+ st x <> {} holds ex z being Element of RAT+ st y = x *' z
t56_arytm_3:: for x, y, z being Element of RAT+ st x <> {} & x *' y = x *' z holds y = z
t57_arytm_3:: for x, y, z being Element of RAT+ holds x *' (y + z) = (x *' y) + (x *' z)
t58_arytm_3:: for i, j being ordinal Element of RAT+ holds i + j = i +^ j
t59_arytm_3:: for i, j being ordinal Element of RAT+ holds i *' j = i *^ j
t6_arytm_3:: for m, n being natural Ordinal st {} in m holds n mod^ m in m
t60_arytm_3:: for x being Element of RAT+ ex y being Element of RAT+ st x = y + y
t61_arytm_3:: for y being set holds not [{},y] in RAT+
t62_arytm_3:: for s, t, r being Element of RAT+ st s + t = r + t holds s = r
t63_arytm_3:: for r, s being Element of RAT+ st r + s = {} holds r = {}
t64_arytm_3:: for s being Element of RAT+ holds {} <=' s
t65_arytm_3:: for s being Element of RAT+ st s <=' {} holds s = {}
t66_arytm_3:: for r, s being Element of RAT+ st r <=' s & s <=' r holds r = s
t67_arytm_3:: for r, s, t being Element of RAT+ st r <=' s & s <=' t holds r <=' t
t68_arytm_3:: for r, s being Element of RAT+ holds ( r < s iff ( r <=' s & r <> s ) ) by Th66;
t69_arytm_3:: for r, s, t being Element of RAT+ st ( ( r < s & s <=' t ) or ( r <=' s & s < t ) ) holds r < t by Th67;
t7_arytm_3:: for n, m being natural Ordinal holds ( m divides n iff n = m *^ (n div^ m) )
t70_arytm_3:: for r, s, t being Element of RAT+ st r < s & s < t holds r < t by Th67;
t71_arytm_3:: for x, y being Element of RAT+ st x in omega & x + y in omega holds y in omega
t72_arytm_3:: for x being Element of RAT+ for i being ordinal Element of RAT+ st i < x & x < i + one holds not x in omega
t73_arytm_3:: for t being Element of RAT+ st t <> {} holds ex r being Element of RAT+ st ( r < t & not r in omega )
t74_arytm_3:: for t being Element of RAT+ holds ( { s where s is Element of RAT+ : s < t } in RAT+ iff t = {} )
t75_arytm_3:: for A being Subset of RAT+ st ex t being Element of RAT+ st ( t in A & t <> {} ) & ( for r, s being Element of RAT+ st r in A & s <=' r holds s in A ) holds ex r1, r2, r3 being Element of RAT+ st ( r1 in A & r2 in A & r3 in A & r1 <> r2 & r2 <> r3 & r3 <> r1 )
t76_arytm_3:: for s, t, r being Element of RAT+ holds ( s + t <=' r + t iff s <=' r )
t77_arytm_3:: for s, t being Element of RAT+ holds s <=' s + t
t78_arytm_3:: for r, s being Element of RAT+ holds ( not r *' s = {} or r = {} or s = {} )
t79_arytm_3:: for r, s, t being Element of RAT+ st r <=' s *' t holds ex t0 being Element of RAT+ st ( r = s *' t0 & t0 <=' t )
t8_arytm_3:: for n, m being natural Ordinal st n divides m & m divides n holds n = m
t80_arytm_3:: for t, s, r being Element of RAT+ st t <> {} & s *' t <=' r *' t holds s <=' r
t81_arytm_3:: for r1, r2, s1, s2 being Element of RAT+ holds ( not r1 + r2 = s1 + s2 or r1 <=' s1 or r2 <=' s2 )
t82_arytm_3:: for s, r, t being Element of RAT+ st s <=' r holds s *' t <=' r *' t
t83_arytm_3:: for r1, r2, s1, s2 being Element of RAT+ holds ( not r1 *' r2 = s1 *' s2 or r1 <=' s1 or r2 <=' s2 )
t84_arytm_3:: for r, s being Element of RAT+ holds ( r = {} iff r + s = s )
t85_arytm_3:: for s1, t1, s2, t2 being Element of RAT+ st s1 + t1 = s2 + t2 & s1 <=' s2 holds t2 <=' t1
t86_arytm_3:: for r, s, t being Element of RAT+ st r <=' s & s <=' r + t holds ex t0 being Element of RAT+ st ( s = r + t0 & t0 <=' t )
t87_arytm_3:: for r, s, t being Element of RAT+ st r <=' s + t holds ex s0, t0 being Element of RAT+ st ( r = s0 + t0 & s0 <=' s & t0 <=' t )
t88_arytm_3:: for r, s, t being Element of RAT+ st r < s & r < t holds ex t0 being Element of RAT+ st ( t0 <=' s & t0 <=' t & r < t0 )
t89_arytm_3:: for r, s, t being Element of RAT+ st r <=' s & s <=' t & s <> t holds r <> t by Th66;
t9_arytm_3:: for n being natural Ordinal holds ( n divides {} & 1 divides n )
t90_arytm_3:: for s, r, t being Element of RAT+ st s < r + t & t <> {} holds ex r0, t0 being Element of RAT+ st ( s = r0 + t0 & r0 <=' r & t0 <=' t & t0 <> t )
t91_arytm_3:: for A being non empty Subset of RAT+ st A in RAT+ holds ex s being Element of RAT+ st ( s in A & ( for r being Element of RAT+ st r in A holds r <=' s ) )
t92_arytm_3:: for r, s being Element of RAT+ ex t being Element of RAT+ st ( r + t = s or s + t = r )
t93_arytm_3:: for r, s being Element of RAT+ st r < s holds ex t being Element of RAT+ st ( r < t & t < s )
t94_arytm_3:: for r being Element of RAT+ ex s being Element of RAT+ st r < s
t95_arytm_3:: for t, r being Element of RAT+ st t <> {} holds ex s being Element of RAT+ st ( s in omega & r <=' s *' t )
d1_asympt_0:: for c being real number holds ( c is logbase iff ( c > 0 & c <> 1 ) );
d10_asympt_0:: for f being eventually-nonnegative Real_Sequence holds Big_Omega f = { t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st ( d > 0 & ( for n being Element of NAT st n >= N holds ( t . n >= d * (f . n) & t . n >= 0 ) ) ) } ;
d11_asympt_0:: for f being eventually-nonnegative Real_Sequence holds Big_Theta f = (Big_Oh f) /\ (Big_Omega f);
d12_asympt_0:: for f being eventually-nonnegative Real_Sequence for X being set holds Big_Oh (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st ( c > 0 & ( for n being Element of NAT st n >= N & n in X holds ( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;
d13_asympt_0:: for f being eventually-nonnegative Real_Sequence for X being set holds Big_Omega (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex d being Real ex N being Element of NAT st ( d > 0 & ( for n being Element of NAT st n >= N & n in X holds ( t . n >= d * (f . n) & t . n >= 0 ) ) ) } ;
d14_asympt_0:: for f being eventually-nonnegative Real_Sequence for X being set holds Big_Theta (f,X) = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st ( c > 0 & d > 0 & ( for n being Element of NAT st n >= N & n in X holds ( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) } ;
d15_asympt_0:: for f being Real_Sequence for b being Element of NAT for b3 being Real_Sequence holds ( b3 = f taken_every b iff for n being Element of NAT holds b3 . n = f . (b * n) );
d16_asympt_0:: for f being eventually-nonnegative Real_Sequence for b being Element of NAT holds ( f is_smooth_wrt b iff ( f is eventually-nondecreasing & f taken_every b in Big_Oh f ) );
d17_asympt_0:: for f being eventually-nonnegative Real_Sequence holds ( f is smooth iff for b being Element of NAT st b >= 2 holds f is_smooth_wrt b );
d18_asympt_0:: for X being non empty set for F, G being FUNCTION_DOMAIN of X, REAL holds F + G = { t where t is Element of Funcs (X,REAL) : ex f, g being Element of Funcs (X,REAL) st ( f in F & g in G & ( for n being Element of X holds t . n = (f . n) + (g . n) ) ) } ;
d19_asympt_0:: for X being non empty set for F, G being FUNCTION_DOMAIN of X, REAL holds max (F,G) = { t where t is Element of Funcs (X,REAL) : ex f, g being Element of Funcs (X,REAL) st ( f in F & g in G & ( for n being Element of X holds t . n = max ((f . n),(g . n)) ) ) } ;
d2_asympt_0:: for f being Real_Sequence holds ( f is eventually-nonnegative iff ex N being Element of NAT st for n being Element of NAT st n >= N holds f . n >= 0 );
d20_asympt_0:: for F, G being FUNCTION_DOMAIN of NAT , REAL holds F to_power G = { t where t is Element of Funcs (NAT,REAL) : ex f, g being Element of Funcs (NAT,REAL) ex N being Element of NAT st ( f in F & g in G & ( for n being Element of NAT st n >= N holds t . n = (f . n) to_power (g . n) ) ) } ;
d3_asympt_0:: for f being Real_Sequence holds ( f is positive iff for n being Element of NAT holds f . n > 0 );
d4_asympt_0:: for f being Real_Sequence holds ( f is eventually-positive iff ex N being Element of NAT st for n being Element of NAT st n >= N holds f . n > 0 );
d5_asympt_0:: for f being Real_Sequence holds ( f is eventually-nonzero iff ex N being Element of NAT st for n being Element of NAT st n >= N holds f . n <> 0 );
d6_asympt_0:: for f being Real_Sequence holds ( f is eventually-nondecreasing iff ex N being Element of NAT st for n being Element of NAT st n >= N holds f . n <= f . (n + 1) );
d7_asympt_0:: for f, g, b3 being Real_Sequence holds ( b3 = max (f,g) iff for n being Element of NAT holds b3 . n = max ((f . n),(g . n)) );
d8_asympt_0:: for f, g being Real_Sequence holds ( g majorizes f iff ex N being Element of NAT st for n being Element of NAT st n >= N holds f . n <= g . n );
d9_asympt_0:: for f being eventually-nonnegative Real_Sequence holds Big_Oh f = { t where t is Element of Funcs (NAT,REAL) : ex c being Real ex N being Element of NAT st ( c > 0 & ( for n being Element of NAT st n >= N holds ( t . n <= c * (f . n) & t . n >= 0 ) ) ) } ;
s1_asympt_0:: scheme FinSegRng1{ F1() -> Nat, F2() -> Nat, F3() -> non empty set , F4( set ) -> Element of F3() } : { F4(i) where i is Element of NAT : ( F1() <= i & i <= F2() ) } is non empty finite Subset of F3() provided A1: F1() <= F2()
s2_asympt_0:: scheme FinImInit1{ F1() -> Nat, F2() -> non empty set , F3( set ) -> Element of F2() } : { F3(n) where n is Element of NAT : n <= F1() } is non empty finite Subset of F2()
s3_asympt_0:: scheme FinImInit2{ F1() -> Nat, F2() -> non empty set , F3( set ) -> Element of F2() } : { F3(n) where n is Element of NAT : n < F1() } is non empty finite Subset of F2() provided A1: F1() > 0
t1_asympt_0:: for f being Real_Sequence for N being Element of NAT st ( for n being Element of NAT st n >= N holds f . n <= f . (n + 1) ) holds for n, m being Element of NAT st N <= n & n <= m holds f . n <= f . m
t10_asympt_0:: for f being eventually-nonnegative Real_Sequence holds f in Big_Oh f
t11_asympt_0:: for f, g being eventually-nonnegative Real_Sequence st f in Big_Oh g holds Big_Oh f c= Big_Oh g
t12_asympt_0:: for f, g, h being eventually-nonnegative Real_Sequence st f in Big_Oh g & g in Big_Oh h holds f in Big_Oh h
t13_asympt_0:: for f being eventually-nonnegative Real_Sequence for c being positive Real holds Big_Oh f = Big_Oh (c (#) f)
t14_asympt_0:: for c being non negative Real for x, f being eventually-nonnegative Real_Sequence st x in Big_Oh f holds x in Big_Oh (c + f)
t15_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds Big_Oh f = Big_Oh g
t16_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) = 0 holds ( f in Big_Oh g & not g in Big_Oh f )
t17_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is divergent_to+infty holds ( not f in Big_Oh g & g in Big_Oh f )
t18_asympt_0:: for x being set for f being eventually-nonnegative Real_Sequence st x in Big_Omega f holds x is eventually-nonnegative Real_Sequence
t19_asympt_0:: for f, g being eventually-nonnegative Real_Sequence holds ( f in Big_Omega g iff g in Big_Oh f )
t2_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) <> 0 holds ( g /" f is convergent & lim (g /" f) = (lim (f /" g)) " )
t20_asympt_0:: for f being eventually-nonnegative Real_Sequence holds f in Big_Omega f
t21_asympt_0:: for f, g, h being eventually-nonnegative Real_Sequence st f in Big_Omega g & g in Big_Omega h holds f in Big_Omega h
t22_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds Big_Omega f = Big_Omega g
t23_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) = 0 holds ( g in Big_Omega f & not f in Big_Omega g )
t24_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is divergent_to+infty holds ( not g in Big_Omega f & f in Big_Omega g )
t25_asympt_0:: for f, t being positive Real_Sequence holds ( t in Big_Omega f iff ex d being Real st ( d > 0 & ( for n being Element of NAT holds d * (f . n) <= t . n ) ) )
t26_asympt_0:: for f, g being eventually-nonnegative Real_Sequence holds Big_Omega (f + g) = Big_Omega (max (f,g))
t27_asympt_0:: for f being eventually-nonnegative Real_Sequence holds Big_Theta f = { t where t is Element of Funcs (NAT,REAL) : ex c, d being Real ex N being Element of NAT st ( c > 0 & d > 0 & ( for n being Element of NAT st n >= N holds ( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) }
t28_asympt_0:: for f being eventually-nonnegative Real_Sequence holds f in Big_Theta f
t29_asympt_0:: for f, g being eventually-nonnegative Real_Sequence st f in Big_Theta g holds g in Big_Theta f
t3_asympt_0:: for f being eventually-nonnegative Real_Sequence st f is convergent holds 0 <= lim f
t30_asympt_0:: for f, g, h being eventually-nonnegative Real_Sequence st f in Big_Theta g & g in Big_Theta h holds f in Big_Theta h
t31_asympt_0:: for f, t being positive Real_Sequence holds ( t in Big_Theta f iff ex c, d being Real st ( c > 0 & d > 0 & ( for n being Element of NAT holds ( d * (f . n) <= t . n & t . n <= c * (f . n) ) ) ) )
t32_asympt_0:: for f, g being eventually-nonnegative Real_Sequence holds Big_Theta (f + g) = Big_Theta (max (f,g))
t33_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) > 0 holds f in Big_Theta g
t34_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is convergent & lim (f /" g) = 0 holds ( f in Big_Oh g & not f in Big_Theta g )
t35_asympt_0:: for f, g being eventually-positive Real_Sequence st f /" g is divergent_to+infty holds ( f in Big_Omega g & not f in Big_Theta g )
t36_asympt_0:: for f being eventually-nonnegative Real_Sequence for X being set holds Big_Theta (f,X) = (Big_Oh (f,X)) /\ (Big_Omega (f,X))
t37_asympt_0:: for f being eventually-nonnegative Real_Sequence st ex b being Element of NAT st ( b >= 2 & f is_smooth_wrt b ) holds f is smooth
t38_asympt_0:: for f being eventually-nonnegative Real_Sequence for t being eventually-nonnegative eventually-nondecreasing Real_Sequence for b being Element of NAT st f is smooth & b >= 2 & t in Big_Oh (f, { (b |^ n) where n is Element of NAT : verum } ) holds t in Big_Oh f
t39_asympt_0:: for f being eventually-nonnegative Real_Sequence for t being eventually-nonnegative eventually-nondecreasing Real_Sequence for b being Element of NAT st f is smooth & b >= 2 & t in Big_Omega (f, { (b |^ n) where n is Element of NAT : verum } ) holds t in Big_Omega f
t4_asympt_0:: for f, g being Real_Sequence st f is convergent & g is convergent & g majorizes f holds lim f <= lim g
t40_asympt_0:: for f being eventually-nonnegative Real_Sequence for t being eventually-nonnegative eventually-nondecreasing Real_Sequence for b being Element of NAT st f is smooth & b >= 2 & t in Big_Theta (f, { (b |^ n) where n is Element of NAT : verum } ) holds t in Big_Theta f
t41_asympt_0:: for f, g being eventually-nonnegative Real_Sequence holds (Big_Oh f) + (Big_Oh g) = Big_Oh (f + g)
t42_asympt_0:: for f, g being eventually-nonnegative Real_Sequence holds max ((Big_Oh f),(Big_Oh g)) = Big_Oh (max (f,g))
t5_asympt_0:: for f being Real_Sequence for g being eventually-nonzero Real_Sequence st f /" g is divergent_to+infty holds ( g /" f is convergent & lim (g /" f) = 0 )
t6_asympt_0:: for x being set for f being eventually-nonnegative Real_Sequence st x in Big_Oh f holds x is eventually-nonnegative Real_Sequence
t7_asympt_0:: for f being positive Real_Sequence for t being eventually-nonnegative Real_Sequence holds ( t in Big_Oh f iff ex c being Real st ( c > 0 & ( for n being Element of NAT holds t . n <= c * (f . n) ) ) )
t8_asympt_0:: for f being eventually-positive Real_Sequence for t being eventually-nonnegative Real_Sequence for N being Element of NAT st t in Big_Oh f & ( for n being Element of NAT st n >= N holds f . n > 0 ) holds ex c being Real st ( c > 0 & ( for n being Element of NAT st n >= N holds t . n <= c * (f . n) ) )
t9_asympt_0:: for f, g being eventually-nonnegative Real_Sequence holds Big_Oh (f + g) = Big_Oh (max (f,g))
d1_asympt_1:: for a, b, c being Real for b4 being Real_Sequence holds ( b4 = seq_a^ (a,b,c) iff for n being Element of NAT holds b4 . n = a to_power ((b * n) + c) );
d2_asympt_1:: for b1 being Real_Sequence holds ( b1 = seq_logn iff ( b1 . 0 = 0 & ( for n being Element of NAT st n > 0 holds b1 . n = log (2,n) ) ) );
d3_asympt_1:: for a being Real for b2 being Real_Sequence holds ( b2 = seq_n^ a iff ( b2 . 0 = 0 & ( for n being Element of NAT st n > 0 holds b2 . n = n to_power a ) ) );
d4_asympt_1:: for b being Real holds seq_const b = NAT --> b;
d5_asympt_1:: for a being Element of NAT for b2 being Real_Sequence holds ( b2 = seq_n! a iff for n being Element of NAT holds b2 . n = (n + a) ! );
d6_asympt_1:: POWEROF2SET = { (2 to_power n) where n is Element of NAT : verum } ;
d7_asympt_1:: for x, b2 being Element of NAT holds ( ( x <> 0 implies ( b2 = Step1 x iff ex n being Element of NAT st ( n ! <= x & x < (n + 1) ! & b2 = n ! ) ) ) & ( not x <> 0 implies ( b2 = Step1 x iff b2 = 0 ) ) );
t1_asympt_1:: for t, t1 being Real_Sequence st t . 0 = 0 & ( for n being Element of NAT st n > 0 holds t . n = ((((12 * (n to_power 3)) * (log (2,n))) - (5 * (n ^2))) + ((log (2,n)) ^2)) + 36 ) & ( for n being Element of NAT st n > 0 holds t1 . n = (n to_power 3) * (log (2,n)) ) holds ex s, s1 being eventually-positive Real_Sequence st ( s = t & s1 = t1 & s in Big_Oh s1 )
t10_asympt_1:: for f being Real_Sequence st ( for n being Element of NAT holds f . n = ((3 * (10 to_power 6)) - ((18 * (10 to_power 3)) * n)) + (27 * (n ^2)) ) holds f in Big_Oh (seq_n^ 2)
t11_asympt_1:: seq_n^ 2 in Big_Oh (seq_n^ 3)
t12_asympt_1:: not seq_n^ 2 in Big_Omega (seq_n^ 3)
t13_asympt_1:: ex s being eventually-positive Real_Sequence st ( s = seq_a^ (2,1,1) & seq_a^ (2,1,0) in Big_Theta s )
t14_asympt_1:: not seq_n! 0 in Big_Theta (seq_n! 1)
t15_asympt_1:: for f being Real_Sequence st f in Big_Oh (seq_n^ 1) holds f (#) f in Big_Oh (seq_n^ 2)
t16_asympt_1:: ex s being eventually-positive Real_Sequence st ( s = seq_a^ (2,1,0) & 2 (#) (seq_n^ 1) in Big_Oh (seq_n^ 1) & not seq_a^ (2,2,0) in Big_Oh s )
t17_asympt_1:: ( log (2,3) < 159 / 100 implies ( seq_n^ (log (2,3)) in Big_Oh (seq_n^ (159 / 100)) & not seq_n^ (log (2,3)) in Big_Omega (seq_n^ (159 / 100)) & not seq_n^ (log (2,3)) in Big_Theta (seq_n^ (159 / 100)) ) )
t18_asympt_1:: for f, g being Real_Sequence st ( for n being Element of NAT holds f . n = n mod 2 ) & ( for n being Element of NAT holds g . n = (n + 1) mod 2 ) holds ex s, s1 being eventually-nonnegative Real_Sequence st ( s = f & s1 = g & not s in Big_Oh s1 & not s1 in Big_Oh s )
t19_asympt_1:: for f, g being eventually-nonnegative Real_Sequence holds ( Big_Oh f = Big_Oh g iff f in Big_Theta g )
t2_asympt_1:: for a, b being logbase Real for f, g being Real_Sequence st a > 1 & b > 1 & ( for n being Element of NAT st n > 0 holds f . n = log (a,n) ) & ( for n being Element of NAT st n > 0 holds g . n = log (b,n) ) holds ex s, s1 being eventually-positive Real_Sequence st ( s = f & s1 = g & Big_Oh s = Big_Oh s1 )
t20_asympt_1:: for f, g being eventually-nonnegative Real_Sequence holds ( f in Big_Theta g iff Big_Theta f = Big_Theta g )
t21_asympt_1:: for e being Real for f being Real_Sequence st 0 < e & ( for n being Element of NAT st n > 0 holds f . n = n * (log (2,n)) ) holds ex s being eventually-positive Real_Sequence st ( s = f & Big_Oh s c= Big_Oh (seq_n^ (1 + e)) & not Big_Oh s = Big_Oh (seq_n^ (1 + e)) )
t22_asympt_1:: for e being Real for g being Real_Sequence st e < 1 & ( for n being Element of NAT st n > 1 holds g . n = (n to_power 2) / (log (2,n)) ) holds ex s being eventually-positive Real_Sequence st ( s = g & Big_Oh (seq_n^ (1 + e)) c= Big_Oh s & not Big_Oh (seq_n^ (1 + e)) = Big_Oh s )
t23_asympt_1:: for f being Real_Sequence st ( for n being Element of NAT st n > 1 holds f . n = (n to_power 2) / (log (2,n)) ) holds ex s being eventually-positive Real_Sequence st ( s = f & Big_Oh s c= Big_Oh (seq_n^ 8) & not Big_Oh s = Big_Oh (seq_n^ 8) )
t24_asympt_1:: for g being Real_Sequence st ( for n being Element of NAT holds g . n = (((n ^2) - n) + 1) to_power 4 ) holds ex s being eventually-positive Real_Sequence st ( s = g & Big_Oh (seq_n^ 8) = Big_Oh s )
t25_asympt_1:: for e being Real st 0 < e & e < 1 holds ex s being eventually-positive Real_Sequence st ( s = seq_a^ ((1 + e),1,0) & Big_Oh (seq_n^ 8) c= Big_Oh s & not Big_Oh (seq_n^ 8) = Big_Oh s )
t26_asympt_1:: for f, g being Real_Sequence st ( for n being Element of NAT st n > 0 holds f . n = n to_power (log (2,n)) ) & ( for n being Element of NAT st n > 0 holds g . n = n to_power (sqrt n) ) holds ex s, s1 being eventually-positive Real_Sequence st ( s = f & s1 = g & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
t27_asympt_1:: for f being Real_Sequence st ( for n being Element of NAT st n > 0 holds f . n = n to_power (sqrt n) ) holds ex s, s1 being eventually-positive Real_Sequence st ( s = f & s1 = seq_a^ (2,1,0) & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
t28_asympt_1:: ex s, s1 being eventually-positive Real_Sequence st ( s = seq_a^ (2,1,0) & s1 = seq_a^ (2,1,1) & Big_Oh s = Big_Oh s1 )
t29_asympt_1:: ex s, s1 being eventually-positive Real_Sequence st ( s = seq_a^ (2,1,0) & s1 = seq_a^ (2,2,0) & Big_Oh s c= Big_Oh s1 & not Big_Oh s = Big_Oh s1 )
t3_asympt_1:: for a, b being positive Real st a < b holds not seq_a^ (b,1,0) in Big_Oh (seq_a^ (a,1,0))
t30_asympt_1:: ex s being eventually-positive Real_Sequence st ( s = seq_a^ (2,2,0) & Big_Oh s c= Big_Oh (seq_n! 0) & not Big_Oh s = Big_Oh (seq_n! 0) )
t31_asympt_1:: ( Big_Oh (seq_n! 0) c= Big_Oh (seq_n! 1) & not Big_Oh (seq_n! 0) = Big_Oh (seq_n! 1) )
t32_asympt_1:: for g being Real_Sequence st ( for n being Element of NAT st n > 0 holds g . n = n to_power n ) holds ex s being eventually-positive Real_Sequence st ( s = g & Big_Oh (seq_n! 1) c= Big_Oh s & not Big_Oh (seq_n! 1) = Big_Oh s )
t33_asympt_1:: for n being Element of NAT st n >= 1 holds for f being Real_Sequence for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds f . n >= (n to_power (k + 1)) / (k + 1)
t34_asympt_1:: for f, g being Real_Sequence st ( for n being Element of NAT st n > 0 holds g . n = n * (log (2,n)) ) & ( for n being Element of NAT holds f . n = log (2,(n !)) ) holds ex s being eventually-nonnegative Real_Sequence st ( s = g & f in Big_Theta s )
t35_asympt_1:: for f being eventually-nonnegative eventually-nondecreasing Real_Sequence for t being Real_Sequence st ( for n being Element of NAT holds ( ( n mod 2 = 0 implies t . n = 1 ) & ( n mod 2 = 1 implies t . n = n ) ) ) holds not t in Big_Theta f
t36_asympt_1:: for f being Real_Sequence st ( for n being Element of NAT holds ( ( n in POWEROF2SET implies f . n = n ) & ( not n in POWEROF2SET implies f . n = 2 to_power n ) ) ) holds ( f in Big_Theta ((seq_n^ 1),POWEROF2SET) & not f in Big_Theta (seq_n^ 1) & seq_n^ 1 is smooth & not f is eventually-nondecreasing )
t37_asympt_1:: for f, g being Real_Sequence st ( for n being Element of NAT st n > 0 holds f . n = n to_power (2 to_power [\(log (2,n))/]) ) & ( for n being Element of NAT st n > 0 holds g . n = n to_power n ) holds ex s being eventually-positive Real_Sequence st ( s = g & f in Big_Theta (s,POWEROF2SET) & not f in Big_Theta s & f is eventually-nondecreasing & s is eventually-nondecreasing & not s is_smooth_wrt 2 )
t38_asympt_1:: for g being Real_Sequence st ( for n being Element of NAT holds ( ( n in POWEROF2SET implies g . n = n ) & ( not n in POWEROF2SET implies g . n = n to_power 2 ) ) ) holds ex s being eventually-positive Real_Sequence st ( s = g & seq_n^ 1 in Big_Theta (s,POWEROF2SET) & not seq_n^ 1 in Big_Theta s & s taken_every 2 in Big_Oh s & seq_n^ 1 is eventually-nondecreasing & not s is eventually-nondecreasing )
t39_asympt_1:: for f being Real_Sequence st ( for n being Element of NAT holds f . n = Step1 n ) holds ex s being eventually-positive Real_Sequence st ( s = f & f is eventually-nondecreasing & ( for n being Element of NAT holds f . n <= (seq_n^ 1) . n ) & not s is smooth )
t4_asympt_1:: for f, g being eventually-nonnegative Real_Sequence holds ( Big_Oh f c= Big_Oh g & not Big_Oh f = Big_Oh g iff ( f in Big_Oh g & not f in Big_Omega g ) )
t40_asympt_1:: for F being eventually-nonnegative Real_Sequence st F = (seq_n^ 1) - (seq_const 1) holds (Big_Theta F) + (Big_Theta (seq_n^ 1)) = Big_Theta (seq_n^ 1)
t41_asympt_1:: ex F being FUNCTION_DOMAIN of NAT , REAL st ( F = {(seq_n^ 1)} & ( for n being Element of NAT holds (seq_n^ (- 1)) . n <= (seq_n^ 1) . n ) & not seq_n^ (- 1) in F to_power (Big_Oh (seq_const 1)) )
t42_asympt_1:: for c being non negative Real for x, f being eventually-nonnegative Real_Sequence st ex e being Real ex N being Element of NAT st ( e > 0 & ( for n being Element of NAT st n >= N holds f . n >= e ) ) & x in Big_Oh (c + f) holds x in Big_Oh f
t43_asympt_1:: 2 to_power 12 = 4096 by Lm26;
t44_asympt_1:: for n being Element of NAT st n >= 3 holds n ^2 > (2 * n) + 1 by Lm27;
t45_asympt_1:: for n being Element of NAT st n >= 10 holds 2 to_power (n - 1) > (2 * n) ^2 by Lm28;
t46_asympt_1:: for n being Element of NAT st n >= 9 holds (n + 1) to_power 6 < 2 * (n to_power 6) by Lm29;
t47_asympt_1:: for n being Element of NAT st n >= 30 holds 2 to_power n > n to_power 6 by Lm30;
t48_asympt_1:: for x being Real st x > 9 holds 2 to_power x > (2 * x) ^2 by Lm31;
t49_asympt_1:: ex N being Element of NAT st for n being Element of NAT st n >= N holds (sqrt n) - (log (2,n)) > 1 by Lm32;
t5_asympt_1:: ( Big_Oh seq_logn c= Big_Oh (seq_n^ (1 / 2)) & not Big_Oh seq_logn = Big_Oh (seq_n^ (1 / 2)) )
t50_asympt_1:: for a, b, c being Real st a > 0 & c > 0 & c <> 1 holds a to_power b = c to_power (b * (log (c,a))) by Lm3;
t51_asympt_1:: 5 ! = 120 by Lm33;
t52_asympt_1:: 5 to_power 5 = 3125 by Lm36;
t53_asympt_1:: 4 to_power 4 = 256 by Lm37;
t54_asympt_1:: for n being Element of NAT holds ((n ^2) - n) + 1 > 0 by Lm21;
t55_asympt_1:: for n being Element of NAT st n >= 2 holds n ! > 1 by Lm50;
t56_asympt_1:: for n1, n being Element of NAT st n <= n1 holds n ! <= n1 ! by Lm51;
t57_asympt_1:: for k being Element of NAT st k >= 1 holds ex n being Element of NAT st ( n ! <= k & k < (n + 1) ! & ( for m being Element of NAT st m ! <= k & k < (m + 1) ! holds m = n ) ) by Lm52;
t58_asympt_1:: for n being Element of NAT st n >= 2 holds [/(n / 2)\] < n by Lm46;
t59_asympt_1:: for n being Element of NAT st n >= 3 holds n ! > n by Lm53;
t6_asympt_1:: ( seq_n^ (1 / 2) in Big_Omega seq_logn & not seq_logn in Big_Omega (seq_n^ (1 / 2)) )
t60_asympt_1:: (seq_n^ 1) - (seq_const 1) is eventually-positive by Lm54;
t61_asympt_1:: for n being Element of NAT st n >= 2 holds 2 to_power n > n + 1 by Lm1;
t62_asympt_1:: for a being logbase Real for f being Real_Sequence st a > 1 & f . 0 = 0 & ( for n being Element of NAT st n > 0 holds f . n = log (a,n) ) holds f is eventually-positive by Lm2;
t63_asympt_1:: for f, g being eventually-nonnegative Real_Sequence holds ( ( f in Big_Oh g & g in Big_Oh f ) iff Big_Oh f = Big_Oh g ) by Lm5;
t64_asympt_1:: for a, b, c being Real st 0 < a & a <= b & c >= 0 holds a to_power c <= b to_power c by Lm6;
t65_asympt_1:: for n being Element of NAT st n >= 4 holds (2 * n) + 3 < 2 to_power n by Lm7;
t66_asympt_1:: for n being Element of NAT st n >= 6 holds (n + 1) ^2 < 2 to_power n by Lm8;
t67_asympt_1:: for c being Real st c > 6 holds c ^2 < 2 to_power c by Lm9;
t68_asympt_1:: for e being positive Real for f being Real_Sequence st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds f . n = log (2,(n to_power e)) ) holds ( f /" (seq_n^ e) is convergent & lim (f /" (seq_n^ e)) = 0 ) by Lm10;
t69_asympt_1:: for e being Real st e > 0 holds ( seq_logn /" (seq_n^ e) is convergent & lim (seq_logn /" (seq_n^ e)) = 0 ) by Lm11;
t7_asympt_1:: for f being Real_Sequence for k being Element of NAT st ( for n being Element of NAT holds f . n = Sum ((seq_n^ k),n) ) holds f in Big_Theta (seq_n^ (k + 1))
t70_asympt_1:: for f being Real_Sequence for N being Element of NAT st ( for n being Element of NAT st n <= N holds f . n >= 0 ) holds Sum (f,N) >= 0 by Lm12;
t71_asympt_1:: for f, g being Real_Sequence for N being Element of NAT st ( for n being Element of NAT st n <= N holds f . n <= g . n ) holds Sum (f,N) <= Sum (g,N) by Lm13;
t72_asympt_1:: for f being Real_Sequence for b being Real st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds f . n = b ) holds for N being Element of NAT holds Sum (f,N) = b * N by Lm14;
t73_asympt_1:: for f being Real_Sequence for N, M being Element of NAT holds (Sum (f,N,M)) + (f . (N + 1)) = Sum (f,(N + 1),M) by Lm15;
t74_asympt_1:: for f, g being Real_Sequence for M, N being Element of NAT st N >= M + 1 & ( for n being Element of NAT st M + 1 <= n & n <= N holds f . n <= g . n ) holds Sum (f,N,M) <= Sum (g,N,M) by Lm16;
t75_asympt_1:: for n being Element of NAT holds [/(n / 2)\] <= n by Lm17;
t76_asympt_1:: for f being Real_Sequence for b being Real for N being Element of NAT st f . 0 = 0 & ( for n being Element of NAT st n > 0 holds f . n = b ) holds for M being Element of NAT holds Sum (f,N,M) = b * (N - M) by Lm18;
t77_asympt_1:: for f, g being Real_Sequence for N being Element of NAT for c being Real st f is convergent & lim f = c & ( for n being Element of NAT st n >= N holds f . n = g . n ) holds ( g is convergent & lim g = c ) by Lm22;
t78_asympt_1:: for n being Element of NAT st n >= 1 holds ((n ^2) - n) + 1 <= n ^2 by Lm23;
t79_asympt_1:: for n being Element of NAT st n >= 1 holds n ^2 <= 2 * (((n ^2) - n) + 1) by Lm24;
t8_asympt_1:: for f being Real_Sequence st ( for n being Element of NAT st n > 0 holds f . n = n to_power (log (2,n)) ) holds ex s being eventually-positive Real_Sequence st ( s = f & not s is smooth )
t80_asympt_1:: for e being Real st 0 < e & e < 1 holds ex N being Element of NAT st for n being Element of NAT st n >= N holds (n * (log (2,(1 + e)))) - (8 * (log (2,n))) > 8 * (log (2,n)) by Lm25;
t81_asympt_1:: for n being Element of NAT st n >= 10 holds (2 to_power (2 * n)) / (n !) < 1 / (2 to_power (n - 9)) by Lm34;
t82_asympt_1:: for n being Element of NAT st n >= 3 holds 2 * (n - 2) >= n - 1 by Lm35;
t83_asympt_1:: for c being real number st c >= 0 holds c to_power (1 / 2) = sqrt c by Lm39;
t84_asympt_1:: ex N being Element of NAT st for n being Element of NAT st n >= N holds n - ((sqrt n) * (log (2,n))) > n / 2 by Lm40;
t85_asympt_1:: for s being Real_Sequence st ( for n being Element of NAT holds s . n = (1 + (1 / (n + 1))) to_power (n + 1) ) holds s is V41() by Lm41;
t86_asympt_1:: for n being Element of NAT st n >= 1 holds ((n + 1) / n) to_power n <= ((n + 2) / (n + 1)) to_power (n + 1) by Lm42;
t87_asympt_1:: for k, n being Element of NAT st k <= n holds n choose k >= ((n + 1) choose k) / (n + 1) by Lm43;
t88_asympt_1:: for f being Real_Sequence st ( for n being Element of NAT holds f . n = log (2,(n !)) ) holds for n being Element of NAT holds f . n = Sum (seq_logn,n) by Lm44;
t89_asympt_1:: for n being Element of NAT st n >= 4 holds n * (log (2,n)) >= 2 * n by Lm45;
t9_asympt_1:: for f being eventually-nonnegative Real_Sequence ex F being FUNCTION_DOMAIN of NAT , REAL st ( F = {(seq_n^ 1)} & ( f in F to_power (Big_Oh (seq_const 1)) implies ex N being Element of NAT ex c being Real ex k being Element of NAT st ( c > 0 & ( for n being Element of NAT st n >= N holds ( 1 <= f . n & f . n <= c * ((seq_n^ k) . n) ) ) ) ) & ( ex N being Element of NAT ex c being Real ex k being Element of NAT st ( c > 0 & ( for n being Element of NAT st n >= N holds ( 1 <= f . n & f . n <= c * ((seq_n^ k) . n) ) ) ) implies f in F to_power (Big_Oh (seq_const 1)) ) )
t90_asympt_1:: for n being Element of NAT st n >= 2 holds n ^2 > n + 1 by Lm47;
t91_asympt_1:: for n being Element of NAT st n >= 1 holds (2 to_power (n + 1)) - (2 to_power n) > 1 by Lm48;
t92_asympt_1:: for n being Element of NAT st n >= 2 holds not (2 to_power n) - 1 in POWEROF2SET by Lm49;
t93_asympt_1:: for n, k being Element of NAT st k >= 1 & n ! <= k & k < (n + 1) ! holds Step1 k = n ! by Def7;
t94_asympt_1:: for a, b, c being Real st a > 1 & b >= a & c >= 1 holds log (a,c) >= log (b,c) by Lm19;
d1_autalg_1:: for UA being Universal_Algebra for b2 being FUNCTION_DOMAIN of the carrier of UA, the carrier of UA holds ( b2 = UAAut UA iff for h being Function of UA,UA holds ( h in b2 iff h is_isomorphism UA,UA ) );
d2_autalg_1:: for UA being Universal_Algebra for b2 being BinOp of (UAAut UA) holds ( b2 = UAAutComp UA iff for x, y being Element of UAAut UA holds b2 . (x,y) = y * x );
d3_autalg_1:: for UA being Universal_Algebra holds UAAutGroup UA = multMagma(# (UAAut UA),(UAAutComp UA) #);
d4_autalg_1:: for I being set for A, B being ManySortedSet of I st A is_transformable_to B holds MSFuncs (A,B) = product ((Funcs) (A,B));
d5_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S for b3 being MSFunctionSet of U1,U1 holds ( b3 = MSAAut U1 iff for h being ManySortedFunction of U1,U1 holds ( h in b3 iff h is_isomorphism U1,U1 ) );
d6_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S for b3 being BinOp of (MSAAut U1) holds ( b3 = MSAAutComp U1 iff for x, y being Element of MSAAut U1 holds b3 . (x,y) = y ** x );
d7_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S holds MSAAutGroup U1 = multMagma(# (MSAAut U1),(MSAAutComp U1) #);
t1_autalg_1:: for UA being Universal_Algebra holds id the carrier of UA is_isomorphism UA,UA
t10_autalg_1:: for I being set for A, B, C being ManySortedSet of I st A is_transformable_to B & B is_transformable_to C holds A is_transformable_to C
t11_autalg_1:: for x being set for A being ManySortedSet of {x} holds A = x .--> (A . x)
t12_autalg_1:: for I being set for A, B being V2() ManySortedSet of I for F being ManySortedFunction of A,B st F is "1-1" & F is "onto" holds ( F "" is "1-1" & F "" is "onto" )
t13_autalg_1:: for I being set for A, B being V2() ManySortedSet of I for F being ManySortedFunction of A,B st F is "1-1" & F is "onto" holds (F "") "" = F
t14_autalg_1:: for F, G being Function-yielding Function st F is "1-1" & G is "1-1" holds G ** F is "1-1"
t15_autalg_1:: for I being set for A being ManySortedSet of I for B, C being V2() ManySortedSet of I for F being ManySortedFunction of A,B for G being ManySortedFunction of B,C st F is "onto" & G is "onto" holds G ** F is "onto"
t16_autalg_1:: for I being set for A, B, C being V2() ManySortedSet of I for F being ManySortedFunction of A,B for G being ManySortedFunction of B,C st F is "1-1" & F is "onto" & G is "1-1" & G is "onto" holds (G ** F) "" = (F "") ** (G "")
t17_autalg_1:: for I being set for A, B being V2() ManySortedSet of I for F being ManySortedFunction of A,B for G being ManySortedFunction of B,A st F is "1-1" & F is "onto" & G ** F = id A holds G = F ""
t18_autalg_1:: for I being set for A, B being ManySortedSet of I st A is_transformable_to B holds (Funcs) (A,B) is V2()
t19_autalg_1:: for I being set for A, B being ManySortedSet of I st A is_transformable_to B holds for x being set st x in MSFuncs (A,B) holds x is ManySortedFunction of A,B
t2_autalg_1:: for UA being Universal_Algebra holds UAAut UA c= Funcs ( the carrier of UA, the carrier of UA)
t20_autalg_1:: for I being set for A, B being ManySortedSet of I st A is_transformable_to B holds for g being ManySortedFunction of A,B holds g in MSFuncs (A,B)
t21_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S holds id the Sorts of U1 in MSFuncs ( the Sorts of U1, the Sorts of U1) by Th20;
t22_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S for f being Element of MSAAut U1 holds f in MSFuncs ( the Sorts of U1, the Sorts of U1) by Th20;
t23_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S holds MSAAut U1 c= MSFuncs ( the Sorts of U1, the Sorts of U1) ;
t24_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S holds id the Sorts of U1 in MSAAut U1
t25_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S for f being Element of MSAAut U1 holds f "" in MSAAut U1
t26_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S for f1, f2 being Element of MSAAut U1 holds f1 ** f2 in MSAAut U1
t27_autalg_1:: for UA being Universal_Algebra for F being ManySortedFunction of (MSAlg UA),(MSAlg UA) for f being Element of UAAut UA st F = 0 .--> f holds F in MSAAut (MSAlg UA)
t28_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S for x, y being Element of (MSAAutGroup U1) for f, g being Element of MSAAut U1 st x = f & y = g holds x * y = g ** f by Def6;
t29_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S holds id the Sorts of U1 = 1_ (MSAAutGroup U1)
t3_autalg_1:: for UA being Universal_Algebra holds id the carrier of UA in UAAut UA
t30_autalg_1:: for S being non empty non void ManySortedSign for U1 being non-empty MSAlgebra over S for f being Element of MSAAut U1 for g being Element of (MSAAutGroup U1) st f = g holds f "" = g "
t31_autalg_1:: for UA1, UA2 being Universal_Algebra st UA1,UA2 are_similar holds for F being ManySortedFunction of (MSAlg UA1),((MSAlg UA2) Over (MSSign UA1)) holds F . 0 is Function of UA1,UA2
t32_autalg_1:: for UA being Universal_Algebra for f being Element of UAAut UA holds 0 .--> f is ManySortedFunction of (MSAlg UA),(MSAlg UA)
t33_autalg_1:: for UA being Universal_Algebra for h being Function st dom h = UAAut UA & ( for x being set st x in UAAut UA holds h . x = 0 .--> x ) holds h is Homomorphism of (UAAutGroup UA),(MSAAutGroup (MSAlg UA))
t34_autalg_1:: for UA being Universal_Algebra for h being Homomorphism of (UAAutGroup UA),(MSAAutGroup (MSAlg UA)) st ( for x being set st x in UAAut UA holds h . x = 0 .--> x ) holds h is bijective
t35_autalg_1:: for UA being Universal_Algebra holds UAAutGroup UA, MSAAutGroup (MSAlg UA) are_isomorphic
t4_autalg_1:: for UA being Universal_Algebra for f, g being Function of UA,UA st f is Element of UAAut UA & g = f " holds g is_isomorphism UA,UA
t5_autalg_1:: for UA being Universal_Algebra for f being Element of UAAut UA holds f " in UAAut UA
t6_autalg_1:: for UA being Universal_Algebra for f1, f2 being Element of UAAut UA holds f1 * f2 in UAAut UA
t7_autalg_1:: for UA being Universal_Algebra for x, y being Element of (UAAutGroup UA) for f, g being Element of UAAut UA st x = f & y = g holds x * y = g * f by Def2;
t8_autalg_1:: for UA being Universal_Algebra holds id the carrier of UA = 1_ (UAAutGroup UA)
t9_autalg_1:: for UA being Universal_Algebra for f being Element of UAAut UA for g being Element of (UAAutGroup UA) st f = g holds f " = g "
d1_autgroup:: for G being strict Group for b2 being FUNCTION_DOMAIN of the carrier of G, the carrier of G holds ( b2 = Aut G iff ( ( for f being Element of b2 holds f is Homomorphism of G,G ) & ( for h being Homomorphism of G,G holds ( h in b2 iff ( h is one-to-one & h is onto ) ) ) ) );
d2_autgroup:: for G being strict Group for b2 being BinOp of (Aut G) holds ( b2 = AutComp G iff for x, y being Element of Aut G holds b2 . (x,y) = x * y );
d3_autgroup:: for G being strict Group holds AutGroup G = multMagma(# (Aut G),(AutComp G) #);
d4_autgroup:: for G being strict Group for b2 being FUNCTION_DOMAIN of the carrier of G, the carrier of G holds ( b2 = InnAut G iff for f being Element of Funcs ( the carrier of G, the carrier of G) holds ( f in b2 iff ex a being Element of G st for x being Element of G holds f . x = x |^ a ) );
d5_autgroup:: for G being strict Group for b2 being strict normal Subgroup of AutGroup G holds ( b2 = InnAutGroup G iff the carrier of b2 = InnAut G );
d6_autgroup:: for G being strict Group for b being Element of G for b3 being Element of InnAut G holds ( b3 = Conjugate b iff for a being Element of G holds b3 . a = a |^ b );
t1_autgroup:: for G being strict Group for H being Subgroup of G holds ( ( for a, b being Element of G st b is Element of H holds b |^ a in H ) iff H is normal ) by Lm1, Lm2;
t10_autgroup:: for G being strict Group for f being Element of Aut G for g being Element of (AutGroup G) st f = g holds f " = g "
t11_autgroup:: for G being strict Group holds InnAut G c= Funcs ( the carrier of G, the carrier of G)
t12_autgroup:: for G being strict Group for f being Element of InnAut G holds f is Element of Aut G
t13_autgroup:: for G being strict Group holds InnAut G c= Aut G
t14_autgroup:: for G being strict Group for f, g being Element of InnAut G holds (AutComp G) . (f,g) = f * g
t15_autgroup:: for G being strict Group holds id the carrier of G is Element of InnAut G
t16_autgroup:: for G being strict Group for f being Element of InnAut G holds f " is Element of InnAut G
t17_autgroup:: for G being strict Group for f, g being Element of InnAut G holds f * g is Element of InnAut G
t18_autgroup:: for G being strict Group for x, y being Element of (InnAutGroup G) for f, g being Element of InnAut G st x = f & y = g holds x * y = f * g
t19_autgroup:: for G being strict Group holds id the carrier of G = 1_ (InnAutGroup G)
t2_autgroup:: for G being strict Group holds Aut G c= Funcs ( the carrier of G, the carrier of G)
t20_autgroup:: for G being strict Group for f being Element of InnAut G for g being Element of (InnAutGroup G) st f = g holds f " = g "
t21_autgroup:: for G being strict Group for a, b being Element of G holds Conjugate (a * b) = (Conjugate b) * (Conjugate a)
t22_autgroup:: for G being strict Group holds Conjugate (1_ G) = id the carrier of G
t23_autgroup:: for G being strict Group for a being Element of G holds (Conjugate (1_ G)) . a = a
t24_autgroup:: for G being strict Group for a being Element of G holds (Conjugate a) * (Conjugate (a ")) = Conjugate (1_ G)
t25_autgroup:: for G being strict Group for a being Element of G holds (Conjugate (a ")) * (Conjugate a) = Conjugate (1_ G)
t26_autgroup:: for G being strict Group for a being Element of G holds Conjugate (a ") = (Conjugate a) "
t27_autgroup:: for G being strict Group for a being Element of G holds ( (Conjugate a) * (Conjugate (1_ G)) = Conjugate a & (Conjugate (1_ G)) * (Conjugate a) = Conjugate a )
t28_autgroup:: for G being strict Group for f being Element of InnAut G holds ( f * (Conjugate (1_ G)) = f & (Conjugate (1_ G)) * f = f )
t29_autgroup:: for G being strict Group holds InnAutGroup G,G ./. (center G) are_isomorphic
t3_autgroup:: for G being strict Group holds id the carrier of G is Element of Aut G
t30_autgroup:: for G being strict Group st G is commutative Group holds for f being Element of (InnAutGroup G) holds f = 1_ (InnAutGroup G)
t4_autgroup:: for G being strict Group for h being Homomorphism of G,G holds ( h in Aut G iff h is bijective )
t5_autgroup:: for G being strict Group for f being Element of Aut G holds f " is Homomorphism of G,G
t6_autgroup:: for G being strict Group for f being Element of Aut G holds f " is Element of Aut G
t7_autgroup:: for G being strict Group for f1, f2 being Element of Aut G holds f1 * f2 is Element of Aut G
t8_autgroup:: for G being strict Group for x, y being Element of (AutGroup G) for f, g being Element of Aut G st x = f & y = g holds x * y = f * g by Def2;
t9_autgroup:: for G being strict Group holds id the carrier of G = 1_ (AutGroup G)
t1_axioms:: for X, Y being Subset of REAL st ( for x, y being real number st x in X & y in Y holds x <= y ) holds ex z being real number st for x, y being real number st x in X & y in Y holds ( x <= z & z <= y )
t2_axioms:: for x, y being real number st x in NAT & y in NAT holds x + y in NAT
t3_axioms:: for A being Subset of REAL st 0 in A & ( for x being real number st x in A holds x + 1 in A ) holds NAT c= A
t4_axioms:: for k being Nat holds k = { i where i is Element of NAT : i < k }
d1_bagorder:: for n, i, j being Nat for b being ManySortedSet of n for b5 being ManySortedSet of j -' i holds ( b5 = (i,j) -cut b iff for k being Element of NAT st k in j -' i holds b5 . k = b . (i + k) );
d10_bagorder:: for i, n being Nat for o1 being TermOrder of (i + 1) for o2 being TermOrder of (n -' (i + 1)) for b5 being TermOrder of n holds ( b5 = BlockOrder (i,n,o1,o2) iff for p, q being bag of n holds ( [p,q] in b5 iff ( ( (0,(i + 1)) -cut p <> (0,(i + 1)) -cut q & [((0,(i + 1)) -cut p),((0,(i + 1)) -cut q)] in o1 ) or ( (0,(i + 1)) -cut p = (0,(i + 1)) -cut q & [(((i + 1),n) -cut p),(((i + 1),n) -cut q)] in o2 ) ) ) );
d11_bagorder:: for n being Nat for b2 being strict RelStr holds ( b2 = NaivelyOrderedBags n iff ( the carrier of b2 = Bags n & ( for x, y being bag of n holds ( [x,y] in the InternalRel of b2 iff x divides y ) ) ) );
d12_bagorder:: for R being non empty connected Poset for X being Element of Fin the carrier of R st not X is empty holds for b3 being Element of R holds ( b3 = PosetMin X iff ( b3 in X & b3 is_minimal_wrt X, the InternalRel of R ) );
d13_bagorder:: for R being non empty connected Poset for X being Element of Fin the carrier of R st not X is empty holds for b3 being Element of R holds ( b3 = PosetMax X iff ( b3 in X & b3 is_maximal_wrt X, the InternalRel of R ) );
d14_bagorder:: for R being non empty connected Poset for b2 being Function of NAT,(bool [:(Fin the carrier of R),(Fin the carrier of R):]) holds ( b2 = FinOrd-Approx R iff ( dom b2 = NAT & b2 . 0 = { [x,y] where x, y is Element of Fin the carrier of R : ( x = {} or ( x <> {} & y <> {} & PosetMax x <> PosetMax y & [(PosetMax x),(PosetMax y)] in the InternalRel of R ) ) } & ( for n being Nat holds b2 . (n + 1) = { [x,y] where x, y is Element of Fin the carrier of R : ( x <> {} & y <> {} & PosetMax x = PosetMax y & [(x \ {(PosetMax x)}),(y \ {(PosetMax y)})] in b2 . n ) } ) ) );
d15_bagorder:: for R being non empty connected Poset holds FinOrd R = union (rng (FinOrd-Approx R));
d16_bagorder:: for R being non empty connected Poset holds FinPoset R = RelStr(# (Fin the carrier of R),(FinOrd R) #);
d17_bagorder:: for R being non empty connected RelStr for C being non empty set st R is well_founded & C c= the carrier of R holds for b3 being Element of R holds ( b3 = MinElement (C,R) iff ( b3 in C & b3 is_minimal_wrt C, the InternalRel of R ) );
d2_bagorder:: for x being non empty set for n being non empty Element of NAT holds Fin (x,n) = { y where y is Subset of x : ( y is finite & not y is empty & card y c= n ) } ;
d3_bagorder:: for R being non empty RelStr for s being sequence of R holds ( s is non-increasing iff for i being Nat holds [(s . (i + 1)),(s . i)] in the InternalRel of R );
d4_bagorder:: for n being Ordinal for b being bag of n for b3 being Nat holds ( b3 = TotDegree b iff ex f being FinSequence of NAT st ( b3 = Sum f & f = b * (SgmX ((RelIncl n),(support b))) ) );
d5_bagorder:: for n being Ordinal for T being TermOrder of n holds ( T is admissible iff ( T is_strongly_connected_in Bags n & ( for a being bag of n holds [(EmptyBag n),a] in T ) & ( for a, b, c being bag of n st [a,b] in T holds [(a + c),(b + c)] in T ) ) );
d6_bagorder:: for n being Ordinal for b2 being TermOrder of n holds ( b2 = InvLexOrder n iff for p, q being bag of n holds ( [p,q] in b2 iff ( p = q or ex i being Ordinal st ( i in n & p . i < q . i & ( for k being Ordinal st i in k & k in n holds p . k = q . k ) ) ) ) );
d7_bagorder:: for n being Ordinal for o being TermOrder of n st ( for a, b, c being bag of n st [a,b] in o holds [(a + c),(b + c)] in o ) holds for b3 being TermOrder of n holds ( b3 = Graded o iff for a, b being bag of n holds ( [a,b] in b3 iff ( TotDegree a < TotDegree b or ( TotDegree a = TotDegree b & [a,b] in o ) ) ) );
d8_bagorder:: for n being Ordinal holds GrLexOrder n = Graded (LexOrder n);
d9_bagorder:: for n being Ordinal holds GrInvLexOrder n = Graded (InvLexOrder n);
t1_bagorder:: for x, y, z being set st z in x & z in y & x \ {z} = y \ {z} holds x = y
t10_bagorder:: for R being non empty transitive RelStr for s being sequence of R st R is well_founded & s is non-increasing holds ex p being Nat st for r being Nat st p <= r holds s . p = s . r
t11_bagorder:: for X being set for a being Element of X for A being finite Subset of X for R being Order of X st A = {a} & R linearly_orders A holds SgmX (R,A) = <*a*>
t12_bagorder:: for n being Ordinal for b being bag of n for s being finite Subset of n for f, g being FinSequence of NAT st f = b * (SgmX ((RelIncl n),(support b))) & g = b * (SgmX ((RelIncl n),((support b) \/ s))) holds Sum f = Sum g
t13_bagorder:: for n being Ordinal for a, b being bag of n holds TotDegree (a + b) = (TotDegree a) + (TotDegree b)
t14_bagorder:: for n being Ordinal for a, b being bag of n st b divides a holds TotDegree (a -' b) = (TotDegree a) - (TotDegree b)
t15_bagorder:: for n being Ordinal for b being bag of n holds ( TotDegree b = 0 iff b = EmptyBag n )
t16_bagorder:: for i, j, n being Nat holds (i,j) -cut (EmptyBag n) = EmptyBag (j -' i)
t17_bagorder:: for i, j, n being Nat for a, b being bag of n holds (i,j) -cut (a + b) = ((i,j) -cut a) + ((i,j) -cut b)
t18_bagorder:: for X being set holds support (EmptyBag X) = {}
t19_bagorder:: for X being set for b being bag of X st support b = {} holds b = EmptyBag X
t2_bagorder:: for n, k being Element of NAT holds ( k in Seg n iff ( k - 1 is Element of NAT & k - 1 < n ) )
t20_bagorder:: for n, m being Ordinal for b being bag of n st m in n holds b | m is bag of m
t21_bagorder:: for n being Ordinal for a, b being bag of n st b divides a holds support b c= support a
t22_bagorder:: for n being Ordinal holds LexOrder n is admissible
t23_bagorder:: for o being infinite Ordinal holds not LexOrder o is well-ordering
t24_bagorder:: for n being Ordinal holds InvLexOrder n is admissible
t25_bagorder:: for o being Ordinal holds InvLexOrder o is well-ordering
t26_bagorder:: for n being Ordinal for o being TermOrder of n st ( for a, b, c being bag of n st [a,b] in o holds [(a + c),(b + c)] in o ) & o is_strongly_connected_in Bags n holds Graded o is admissible
t27_bagorder:: for n being Ordinal holds GrLexOrder n is admissible
t28_bagorder:: for o being infinite Ordinal holds not GrLexOrder o is well-ordering
t29_bagorder:: for n being Ordinal holds GrInvLexOrder n is admissible
t3_bagorder:: canceled;
t30_bagorder:: for o being Ordinal holds GrInvLexOrder o is well-ordering
t31_bagorder:: for i, n being Nat for o1 being TermOrder of (i + 1) for o2 being TermOrder of (n -' (i + 1)) st o1 is admissible & o2 is admissible holds BlockOrder (i,n,o1,o2) is admissible
t32_bagorder:: for n being Nat holds the carrier of (product (n --> OrderedNAT)) = Bags n
t33_bagorder:: for n being Nat holds NaivelyOrderedBags n = product (n --> OrderedNAT)
t34_bagorder:: for n being Nat for o being TermOrder of n st o is admissible holds ( the InternalRel of (NaivelyOrderedBags n) c= o & o is well-ordering )
t35_bagorder:: for R being non empty connected Poset for x, y being Element of Fin the carrier of R holds ( [x,y] in union (rng (FinOrd-Approx R)) iff ( x = {} or ( x <> {} & y <> {} & PosetMax x <> PosetMax y & [(PosetMax x),(PosetMax y)] in the InternalRel of R ) or ( x <> {} & y <> {} & PosetMax x = PosetMax y & [(x \ {(PosetMax x)}),(y \ {(PosetMax y)})] in union (rng (FinOrd-Approx R)) ) ) )
t36_bagorder:: for R being non empty connected Poset for x being Element of Fin the carrier of R st x <> {} holds not [x,{}] in union (rng (FinOrd-Approx R))
t37_bagorder:: for R being non empty connected Poset for a being Element of Fin the carrier of R holds a \ {(PosetMax a)} is Element of Fin the carrier of R
t38_bagorder:: for R being non empty connected Poset holds union (rng (FinOrd-Approx R)) is Order of (Fin the carrier of R)
t39_bagorder:: for R being non empty connected Poset for a, b being Element of (FinPoset R) holds ( [a,b] in the InternalRel of (FinPoset R) iff ex x, y being Element of Fin the carrier of R st ( a = x & b = y & ( x = {} or ( x <> {} & y <> {} & PosetMax x <> PosetMax y & [(PosetMax x),(PosetMax y)] in the InternalRel of R ) or ( x <> {} & y <> {} & PosetMax x = PosetMax y & [(x \ {(PosetMax x)}),(y \ {(PosetMax y)})] in FinOrd R ) ) ) )
t4_bagorder:: for fs being FinSequence of NAT holds ( Sum fs = 0 iff fs = (len fs) |-> 0 )
t40_bagorder:: for R being non empty RelStr for s being sequence of R for j being Nat st s is descending holds s ^\ j is descending
t41_bagorder:: for R being non empty connected Poset st R is well_founded holds FinPoset R is well_founded
t5_bagorder:: for n, i being Nat for a, b being ManySortedSet of n holds ( a = b iff ( (0,(i + 1)) -cut a = (0,(i + 1)) -cut b & ((i + 1),n) -cut a = ((i + 1),n) -cut b ) )
t6_bagorder:: for R being non empty transitive antisymmetric RelStr for X being finite Subset of R st X <> {} holds ex x being Element of R st ( x in X & x is_maximal_wrt X, the InternalRel of R )
t7_bagorder:: for R being non empty transitive antisymmetric RelStr for X being finite Subset of R st X <> {} holds ex x being Element of R st ( x in X & x is_minimal_wrt X, the InternalRel of R )
t8_bagorder:: for R being non empty transitive antisymmetric RelStr for f being sequence of R st f is descending holds for j, i being Nat st i < j holds ( f . i <> f . j & [(f . j),(f . i)] in the InternalRel of R )
t9_bagorder:: for R being non empty transitive RelStr for f being sequence of R st f is non-increasing holds for j, i being Nat st i < j holds [(f . j),(f . i)] in the InternalRel of R
d1_bcialg_1:: for A being BCIStr for x, y being Element of A holds x \ y = the InternalDiff of A . (x,y);
d10_bcialg_1:: for X, b2 being BCI-algebra holds ( b2 is SubAlgebra of X iff ( 0. b2 = 0. X & the carrier of b2 c= the carrier of X & the InternalDiff of b2 = the InternalDiff of X || the carrier of b2 ) );
d11_bcialg_1:: for IT being non empty BCIStr_0 for x, y being Element of IT holds ( x <= y iff x \ y = 0. IT );
d12_bcialg_1:: for X being BCI-algebra holds BCK-part X = { x where x is Element of X : 0. X <= x } ;
d13_bcialg_1:: for X being BCI-algebra for IT being SubAlgebra of X holds ( IT is proper iff IT <> X );
d14_bcialg_1:: for X being BCI-algebra for IT being Element of X holds ( IT is atom iff for z being Element of X st z \ IT = 0. X holds z = IT );
d15_bcialg_1:: for X being BCI-algebra holds AtomSet X = { x where x is Element of X : x is atom } ;
d16_bcialg_1:: for X being BCI-algebra holds ( X is generated_by_atom iff for x being Element of X ex a being Element of AtomSet X st a <= x );
d17_bcialg_1:: for X being BCI-algebra for a being Element of AtomSet X holds BranchV a = { x where x is Element of X : a <= x } ;
d18_bcialg_1:: for X being BCI-algebra for b2 being non empty Subset of X holds ( b2 is Ideal of X iff ( 0. X in b2 & ( for x, y being Element of X st x \ y in b2 & y in b2 holds x in b2 ) ) );
d19_bcialg_1:: for X being BCI-algebra for IT being Ideal of X holds ( IT is closed iff for x being Element of IT holds x ` in IT );
d2_bcialg_1:: for IT being non empty BCIStr_0 for x being Element of IT holds x ` = (0. IT) \ x;
d20_bcialg_1:: for IT being BCI-algebra holds ( IT is associative iff for x, y, z being Element of IT holds (x \ y) \ z = x \ (y \ z) );
d21_bcialg_1:: for IT being BCI-algebra holds ( IT is quasi-associative iff for x being Element of IT holds (x `) ` = x ` );
d22_bcialg_1:: for IT being BCI-algebra holds ( IT is positive-implicative iff for x, y being Element of IT holds (x \ (x \ y)) \ (y \ x) = x \ (x \ (y \ (y \ x))) );
d23_bcialg_1:: for IT being BCI-algebra holds ( IT is weakly-positive-implicative iff for x, y, z being Element of IT holds (x \ y) \ z = ((x \ z) \ z) \ (y \ z) );
d24_bcialg_1:: for IT being BCI-algebra holds ( IT is implicative iff for x, y being Element of IT holds (x \ (x \ y)) \ (y \ x) = y \ (y \ x) );
d25_bcialg_1:: for IT being BCI-algebra holds ( IT is weakly-implicative iff for x, y being Element of IT holds (x \ (y \ x)) \ ((y \ x) `) = x );
d26_bcialg_1:: for IT being BCI-algebra holds ( IT is p-Semisimple iff for x, y being Element of IT holds x \ (x \ y) = y );
d27_bcialg_1:: for IT being BCI-algebra holds ( IT is alternative iff for x, y being Element of IT holds ( x \ (x \ y) = (x \ x) \ y & (x \ y) \ y = x \ (y \ y) ) );
d3_bcialg_1:: for IT being non empty BCIStr_0 holds ( IT is being_B iff for x, y, z being Element of IT holds ((x \ y) \ (z \ y)) \ (x \ z) = 0. IT );
d4_bcialg_1:: for IT being non empty BCIStr_0 holds ( IT is being_C iff for x, y, z being Element of IT holds ((x \ y) \ z) \ ((x \ z) \ y) = 0. IT );
d5_bcialg_1:: for IT being non empty BCIStr_0 holds ( IT is being_I iff for x being Element of IT holds x \ x = 0. IT );
d6_bcialg_1:: for IT being non empty BCIStr_0 holds ( IT is being_K iff for x, y being Element of IT holds (x \ y) \ x = 0. IT );
d7_bcialg_1:: for IT being non empty BCIStr_0 holds ( IT is being_BCI-4 iff for x, y being Element of IT st x \ y = 0. IT & y \ x = 0. IT holds x = y );
d8_bcialg_1:: for IT being non empty BCIStr_0 holds ( IT is being_BCK-5 iff for x being Element of IT holds x ` = 0. IT );
d9_bcialg_1:: BCI-EXAMPLE = BCIStr_0(# 1,op2,op0 #);
t1_bcialg_1:: for X being non empty BCIStr_0 holds ( X is BCI-algebra iff ( X is being_I & X is being_BCI-4 & ( for x, y, z being Element of X holds ( ((x \ y) \ (x \ z)) \ (z \ y) = 0. X & (x \ (x \ y)) \ y = 0. X ) ) ) )
t10_bcialg_1:: for X being BCI-algebra for x, y being Element of X holds ((x \ (x \ y)) \ (y \ x)) \ (x \ (x \ (y \ (y \ x)))) = 0. X
t11_bcialg_1:: for X being non empty BCIStr_0 holds ( X is BCI-algebra iff ( X is being_BCI-4 & ( for x, y, z being Element of X holds ( ((x \ y) \ (x \ z)) \ (z \ y) = 0. X & x \ (0. X) = x ) ) ) )
t12_bcialg_1:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds x \ (x \ y) = y \ (y \ x) ) holds X is BCK-algebra
t13_bcialg_1:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds (x \ y) \ y = x \ y ) holds X is BCK-algebra
t14_bcialg_1:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds x \ (y \ x) = x ) holds X is BCK-algebra
t15_bcialg_1:: for X being BCI-algebra st ( for X being BCI-algebra for x, y, z being Element of X holds (x \ y) \ y = (x \ z) \ (y \ z) ) holds X is BCK-algebra
t16_bcialg_1:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds (x \ y) \ (y \ x) = x \ y ) holds X is BCK-algebra
t17_bcialg_1:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds (x \ y) \ ((x \ y) \ (y \ x)) = 0. X ) holds X is BCK-algebra
t18_bcialg_1:: for X being BCI-algebra holds ( X is being_K iff X is BCK-algebra )
t19_bcialg_1:: for X being BCI-algebra holds 0. X in BCK-part X
t2_bcialg_1:: for X being BCI-algebra for x being Element of X holds x \ (0. X) = x
t20_bcialg_1:: for X being BCI-algebra for x, y being Element of BCK-part X holds x \ y in BCK-part X
t21_bcialg_1:: for X being BCI-algebra for x being Element of X for y being Element of BCK-part X holds x \ y <= x
t22_bcialg_1:: for X being BCI-algebra holds X is SubAlgebra of X
t23_bcialg_1:: for X being BCI-algebra holds 0. X in AtomSet X
t24_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for z being Element of X holds z \ (z \ x) = x )
t25_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for u, z being Element of X holds (z \ u) \ (z \ x) = x \ u )
t26_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for y, z being Element of X holds x \ (z \ y) <= y \ (z \ x) )
t27_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for y, z, u being Element of X holds (x \ u) \ (z \ y) <= (y \ u) \ (z \ x) )
t28_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for z being Element of X holds (z `) \ (x `) = x \ z )
t29_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff (x `) ` = x )
t3_bcialg_1:: for X being BCI-algebra for x, y, z being Element of X st x \ y = 0. X & y \ z = 0. X holds x \ z = 0. X
t30_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for z being Element of X holds (z \ x) ` = x \ z )
t31_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for z being Element of X holds ((x \ z) `) ` = x \ z )
t32_bcialg_1:: for X being BCI-algebra for x being Element of X holds ( x in AtomSet X iff for z, u being Element of X holds z \ (z \ (x \ u)) = x \ u )
t33_bcialg_1:: for X being BCI-algebra for a being Element of AtomSet X for x being Element of X holds a \ x in AtomSet X
t34_bcialg_1:: for X being BCI-algebra for x being Element of X holds x ` in AtomSet X
t35_bcialg_1:: for X being BCI-algebra for x being Element of X ex a being Element of AtomSet X st a <= x
t36_bcialg_1:: for X being BCI-algebra holds X is generated_by_atom
t37_bcialg_1:: for X being BCI-algebra for a, b being Element of AtomSet X for x being Element of BranchV b holds a \ x = a \ b
t38_bcialg_1:: for X being BCI-algebra for a being Element of AtomSet X for x being Element of BCK-part X holds a \ x = a
t39_bcialg_1:: for X being BCI-algebra for a, b being Element of AtomSet X for x being Element of BranchV a for y being Element of BranchV b holds x \ y in BranchV (a \ b)
t4_bcialg_1:: for X being BCI-algebra for x, y, z being Element of X st x \ y = 0. X holds ( (x \ z) \ (y \ z) = 0. X & (z \ y) \ (z \ x) = 0. X )
t40_bcialg_1:: for X being BCI-algebra for a being Element of AtomSet X for x, y being Element of BranchV a holds x \ y in BCK-part X
t41_bcialg_1:: for X being BCI-algebra for a, b being Element of AtomSet X for x being Element of BranchV a for y being Element of BranchV b st a <> b holds not x \ y in BCK-part X
t42_bcialg_1:: for X being BCI-algebra for a, b being Element of AtomSet X st a <> b holds (BranchV a) /\ (BranchV b) = {}
t43_bcialg_1:: for X being BCI-algebra holds {(0. X)} is closed Ideal of X
t44_bcialg_1:: for X being BCI-algebra holds the carrier of X is closed Ideal of X
t45_bcialg_1:: for X being BCI-algebra holds BCK-part X is closed Ideal of X
t46_bcialg_1:: for X being BCI-algebra for IT being non empty Subset of X st IT is Ideal of X holds for x, y being Element of X st x in IT & y <= x holds y in IT
t47_bcialg_1:: for X being BCI-algebra holds ( X is associative iff for x being Element of X holds x ` = x )
t48_bcialg_1:: for X being BCI-algebra holds ( ( for x, y being Element of X holds y \ x = x \ y ) iff X is associative )
t49_bcialg_1:: for X being non empty BCIStr_0 holds ( X is associative BCI-algebra iff for x, y, z being Element of X holds ( (y \ x) \ (z \ x) = z \ y & x \ (0. X) = x ) )
t5_bcialg_1:: for X being BCI-algebra for x, y, z being Element of X st x <= y holds ( x \ z <= y \ z & z \ y <= z \ x )
t50_bcialg_1:: for X being non empty BCIStr_0 holds ( X is associative BCI-algebra iff for x, y, z being Element of X holds ( (x \ y) \ (x \ z) = z \ y & x ` = x ) )
t51_bcialg_1:: for X being non empty BCIStr_0 holds ( X is associative BCI-algebra iff for x, y, z being Element of X holds ( (x \ y) \ (x \ z) = y \ z & x \ (0. X) = x ) )
t52_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x being Element of X holds x is atom )
t53_bcialg_1:: for X being BCI-algebra st X is p-Semisimple holds BCK-part X = {(0. X)}
t54_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x being Element of X holds (x `) ` = x )
t55_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y being Element of X holds y \ (y \ x) = x )
t56_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z being Element of X holds (z \ y) \ (z \ x) = x \ y )
t57_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z being Element of X holds x \ (z \ y) = y \ (z \ x) )
t58_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z, u being Element of X holds (x \ u) \ (z \ y) = (y \ u) \ (z \ x) )
t59_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, z being Element of X holds (z `) \ (x `) = x \ z )
t6_bcialg_1:: for X being BCI-algebra for x, y being Element of X st x \ y = 0. X holds (y \ x) ` = 0. X
t60_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, z being Element of X holds ((x \ z) `) ` = x \ z )
t61_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, u, z being Element of X holds z \ (z \ (x \ u)) = x \ u )
t62_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x being Element of X st x ` = 0. X holds x = 0. X )
t63_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y being Element of X holds x \ (y `) = y \ (x `) )
t64_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z, u being Element of X holds (x \ y) \ (z \ u) = (x \ z) \ (y \ u) )
t65_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z being Element of X holds (x \ y) \ (z \ y) = x \ z )
t66_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z being Element of X holds x \ (y \ z) = (z \ y) \ (x `) )
t67_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z being Element of X st y \ x = z \ x holds y = z )
t68_bcialg_1:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y, z being Element of X st x \ y = x \ z holds y = z )
t69_bcialg_1:: for X being non empty BCIStr_0 holds ( X is p-Semisimple BCI-algebra iff for x, y, z being Element of X holds ( (x \ y) \ (x \ z) = z \ y & x \ (0. X) = x ) )
t7_bcialg_1:: for X being BCI-algebra for x, y, z being Element of X holds (x \ y) \ z = (x \ z) \ y
t70_bcialg_1:: for X being non empty BCIStr_0 holds ( X is p-Semisimple BCI-algebra iff ( X is being_I & ( for x, y, z being Element of X holds ( x \ (y \ z) = z \ (y \ x) & x \ (0. X) = x ) ) ) )
t71_bcialg_1:: for X being BCI-algebra holds ( X is quasi-associative iff for x being Element of X holds x ` <= x )
t72_bcialg_1:: for X being BCI-algebra holds ( X is quasi-associative iff for x, y being Element of X holds (x \ y) ` = (y \ x) ` )
t73_bcialg_1:: for X being BCI-algebra holds ( X is quasi-associative iff for x, y being Element of X holds (x `) \ y = (x \ y) ` )
t74_bcialg_1:: for X being BCI-algebra holds ( X is quasi-associative iff for x, y being Element of X holds (x \ y) \ (y \ x) in BCK-part X )
t75_bcialg_1:: for X being BCI-algebra holds ( X is quasi-associative iff for x, y, z being Element of X holds (x \ y) \ z <= x \ (y \ z) )
t76_bcialg_1:: for X being BCI-algebra for x, y being Element of X st X is alternative holds ( x ` = x & x \ (x \ y) = y & (x \ y) \ y = x )
t77_bcialg_1:: for X being BCI-algebra for x, a, b being Element of X st X is alternative & x \ a = x \ b holds a = b
t78_bcialg_1:: for X being BCI-algebra for a, x, b being Element of X st X is alternative & a \ x = b \ x holds a = b
t79_bcialg_1:: for X being BCI-algebra for x, y being Element of X st X is alternative & x \ y = 0. X holds x = y
t8_bcialg_1:: for X being BCI-algebra for x, y being Element of X holds x \ (x \ (x \ y)) = x \ y
t80_bcialg_1:: for X being BCI-algebra for x, a, b being Element of X st X is alternative & (x \ a) \ b = 0. X holds ( a = x \ b & b = x \ a )
t81_bcialg_1:: for X being BCI-algebra for x, y being Element of X st X is alternative holds (x \ (x \ y)) \ (y \ x) = x
t82_bcialg_1:: for X being BCI-algebra for y, x being Element of X st X is alternative holds y \ (y \ (x \ (x \ y))) = y
t83_bcialg_1:: for X being non empty BCIStr_0 holds ( X is implicative BCI-algebra iff for x, y, z being Element of X holds ( ((x \ y) \ (x \ z)) \ (z \ y) = 0. X & x \ (0. X) = x & (x \ (x \ y)) \ (y \ x) = y \ (y \ x) ) )
t84_bcialg_1:: for X being BCI-algebra holds ( X is weakly-positive-implicative iff for x, y being Element of X holds x \ y = ((x \ y) \ y) \ (y `) )
t85_bcialg_1:: for X being BCI-algebra st X is weakly-positive-implicative BCI-algebra holds for x, y being Element of X holds (x \ (x \ y)) \ (y \ x) = ((y \ (y \ x)) \ (y \ x)) \ (x \ y) by Lm25;
t86_bcialg_1:: for X being non empty BCIStr_0 holds ( X is positive-implicative BCI-algebra iff for x, y, z being Element of X holds ( ((x \ y) \ (x \ z)) \ (z \ y) = 0. X & x \ (0. X) = x & x \ y = ((x \ y) \ y) \ (y `) & (x \ (x \ y)) \ (y \ x) = ((y \ (y \ x)) \ (y \ x)) \ (x \ y) ) )
t9_bcialg_1:: for X being BCI-algebra for x, y being Element of X holds (x \ y) ` = (x `) \ (y `)
d1_bcialg_2:: for X being BCI-algebra for x, y being Element of X for n being Element of NAT for b5 being Element of X holds ( b5 = (x,y) to_power n iff ex f being Function of NAT, the carrier of X st ( b5 = f . n & f . 0 = x & ( for j being Element of NAT st j < n holds f . (j + 1) = (f . j) \ y ) ) );
d10_bcialg_2:: for X being BCI-algebra for b2 being Equivalence_Relation of X holds ( b2 is L-congruence of X iff for x, y being Element of X st [x,y] in b2 holds for u being Element of X holds [(u \ x),(u \ y)] in b2 );
d11_bcialg_2:: for X being BCI-algebra for b2 being Equivalence_Relation of X holds ( b2 is R-congruence of X iff for x, y being Element of X st [x,y] in b2 holds for u being Element of X holds [(x \ u),(y \ u)] in b2 );
d12_bcialg_2:: for X being BCI-algebra for A being Ideal of X for b3 being Relation of X holds ( b3 is I-congruence of X,A iff for x, y being Element of X holds ( [x,y] in b3 iff ( x \ y in A & y \ x in A ) ) );
d13_bcialg_2:: for X being BCI-algebra for b2 being set holds ( b2 = IConSet X iff for A1 being set holds ( A1 in b2 iff ex I being Ideal of X st A1 is I-congruence of X,I ) );
d14_bcialg_2:: for X being BCI-algebra holds ConSet X = { R where R is Congruence of X : verum } ;
d15_bcialg_2:: for X being BCI-algebra holds LConSet X = { R where R is L-congruence of X : verum } ;
d16_bcialg_2:: for X being BCI-algebra holds RConSet X = { R where R is R-congruence of X : verum } ;
d17_bcialg_2:: for X being BCI-algebra for E being Congruence of X for b3 being BinOp of (Class E) holds ( b3 = EqClaOp E iff for W1, W2 being Element of Class E for x, y being Element of X st W1 = Class (E,x) & W2 = Class (E,y) holds b3 . (W1,W2) = Class (E,(x \ y)) );
d18_bcialg_2:: for X being BCI-algebra for E being Congruence of X holds zeroEqC E = Class (E,(0. X));
d19_bcialg_2:: for X being BCI-algebra for E being Congruence of X holds X ./. E = BCIStr_0(# (Class E),(EqClaOp E),(zeroEqC E) #);
d2_bcialg_2:: for X being BCI-algebra for a being Element of X holds ( a is positive iff 0. X <= a );
d20_bcialg_2:: for X being BCI-algebra for E being Congruence of X for W1, W2 being Element of Class E holds W1 \ W2 = (EqClaOp E) . (W1,W2);
d3_bcialg_2:: for X being BCI-algebra for a being Element of X holds ( a is least iff for x being Element of X holds a <= x );
d4_bcialg_2:: for X being BCI-algebra for a being Element of X holds ( a is maximal iff for x being Element of X st a <= x holds x = a );
d5_bcialg_2:: for X being BCI-algebra for a being Element of X holds ( a is greatest iff for x being Element of X holds x <= a );
d6_bcialg_2:: for X being BCI-algebra for x being Element of X holds ( x is nilpotent iff ex k being non empty Element of NAT st ((0. X),x) to_power k = 0. X );
d7_bcialg_2:: for X being BCI-algebra holds ( X is nilpotent iff for x being Element of X holds x is nilpotent );
d8_bcialg_2:: for X being BCI-algebra for x being Element of X st x is nilpotent holds for b3 being non empty Element of NAT holds ( b3 = ord x iff ( ((0. X),x) to_power b3 = 0. X & ( for m being Element of NAT st ((0. X),x) to_power m = 0. X & m <> 0 holds b3 <= m ) ) );
d9_bcialg_2:: for X being BCI-algebra for b2 being Equivalence_Relation of X holds ( b2 is Congruence of X iff for x, y, u, v being Element of X st [x,y] in b2 & [u,v] in b2 holds [(x \ u),(y \ v)] in b2 );
t1_bcialg_2:: for X being BCI-algebra for x, y being Element of X holds (x,y) to_power 0 = x
t10_bcialg_2:: for X being BCI-algebra for x, y being Element of X for n, m being Element of NAT holds (((x,y) to_power n),y) to_power m = (x,y) to_power (n + m)
t11_bcialg_2:: for X being BCI-algebra for x, y, z being Element of X for n, m being Element of NAT holds (((x,y) to_power n),z) to_power m = (((x,z) to_power m),y) to_power n
t12_bcialg_2:: for X being BCI-algebra for x being Element of X for n being Element of NAT holds ((((0. X),x) to_power n) `) ` = ((0. X),x) to_power n
t13_bcialg_2:: for X being BCI-algebra for x being Element of X for n, m being Element of NAT holds ((0. X),x) to_power (n + m) = (((0. X),x) to_power n) \ ((((0. X),x) to_power m) `)
t14_bcialg_2:: for X being BCI-algebra for x being Element of X for m, n being Element of NAT holds (((0. X),x) to_power (m + n)) ` = ((((0. X),x) to_power m) `) \ (((0. X),x) to_power n)
t15_bcialg_2:: for X being BCI-algebra for x being Element of X for m, n being Element of NAT holds (((0. X),(((0. X),x) to_power m)) to_power n) ` = ((0. X),x) to_power (m * n)
t16_bcialg_2:: for X being BCI-algebra for x being Element of X for m, n being Element of NAT st ((0. X),x) to_power m = 0. X holds ((0. X),x) to_power (m * n) = 0. X
t17_bcialg_2:: for X being BCI-algebra for x, y being Element of X for n being Element of NAT st x \ y = x holds (x,y) to_power n = x
t18_bcialg_2:: for X being BCI-algebra for x, y being Element of X for n being Element of NAT holds ((0. X),(x \ y)) to_power n = (((0. X),x) to_power n) \ (((0. X),y) to_power n)
t19_bcialg_2:: for X being BCI-algebra for x, y, z being Element of X for n being Element of NAT st x <= y holds (x,z) to_power n <= (y,z) to_power n
t2_bcialg_2:: for X being BCI-algebra for x, y being Element of X holds (x,y) to_power 1 = x \ y
t20_bcialg_2:: for X being BCI-algebra for x, y, z being Element of X for n being Element of NAT st x <= y holds (z,y) to_power n <= (z,x) to_power n
t21_bcialg_2:: for X being BCI-algebra for x, z, y being Element of X for n being Element of NAT holds ((x,z) to_power n) \ ((y,z) to_power n) <= x \ y
t22_bcialg_2:: for X being BCI-algebra for x, y being Element of X for n being Element of NAT holds (((x,(x \ y)) to_power n),(y \ x)) to_power n <= x
t23_bcialg_2:: for X being BCI-algebra for a being Element of X holds ( a is minimal iff for x being Element of X holds a \ x = (x `) \ (a `) )
t24_bcialg_2:: for X being BCI-algebra for x being Element of X holds ( x ` is minimal iff for y being Element of X st y <= x holds x ` = y ` )
t25_bcialg_2:: for X being BCI-algebra for x being Element of X holds ( x ` is minimal iff for y, z being Element of X holds (((x \ z) \ (y \ z)) `) ` = (y `) \ (x `) )
t26_bcialg_2:: for X being BCI-algebra st 0. X is maximal holds for a being Element of X holds a is minimal
t27_bcialg_2:: for X being BCI-algebra st ex x being Element of X st x is greatest holds for a being Element of X holds a is positive
t28_bcialg_2:: for X being BCI-algebra for x being Element of X holds x \ ((x `) `) is positive Element of X
t29_bcialg_2:: for X being BCI-algebra for a being Element of X holds ( a is minimal iff (a `) ` = a )
t3_bcialg_2:: for X being BCI-algebra for x, y being Element of X holds (x,y) to_power 2 = (x \ y) \ y
t30_bcialg_2:: for X being BCI-algebra for a being Element of X holds ( a is minimal iff ex x being Element of X st a = x ` )
t31_bcialg_2:: for X being BCI-algebra for x being Element of X holds ( x is positive Element of X iff ( x is nilpotent & ord x = 1 ) )
t32_bcialg_2:: for X being BCI-algebra holds ( X is BCK-algebra iff for x being Element of X holds ( ord x = 1 & x is nilpotent ) )
t33_bcialg_2:: for X being BCI-algebra for x being Element of X for n being Element of NAT holds ((0. X),(x `)) to_power n is minimal
t34_bcialg_2:: for X being BCI-algebra for x being Element of X st x is nilpotent holds ord x = ord (x `)
t35_bcialg_2:: for X being BCI-algebra for E being Congruence of X holds Class (E,(0. X)) is closed Ideal of X
t36_bcialg_2:: for X being BCI-algebra for R being Equivalence_Relation of X holds ( R is Congruence of X iff ( R is R-congruence of X & R is L-congruence of X ) )
t37_bcialg_2:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I holds RI is Congruence of X
t38_bcialg_2:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I holds Class (RI,(0. X)) c= I
t39_bcialg_2:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I holds ( I is closed iff I = Class (RI,(0. X)) )
t4_bcialg_2:: for X being BCI-algebra for x, y being Element of X for n being Element of NAT holds (x,y) to_power (n + 1) = ((x,y) to_power n) \ y
t40_bcialg_2:: for X being BCI-algebra for x, y being Element of X for E being Congruence of X st [x,y] in E holds ( x \ y in Class (E,(0. X)) & y \ x in Class (E,(0. X)) )
t41_bcialg_2:: for X being BCI-algebra for A, I being Ideal of X for IA being I-congruence of X,A for II being I-congruence of X,I st Class (IA,(0. X)) = Class (II,(0. X)) holds IA = II
t42_bcialg_2:: for X being BCI-algebra for x, y, u being Element of X for k being Element of NAT for E being Congruence of X st [x,y] in E & u in Class (E,(0. X)) holds [x,((y,u) to_power k)] in E
t43_bcialg_2:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X ex i, j, m, n being Element of NAT st (((x,(x \ y)) to_power i),(y \ x)) to_power j = (((y,(y \ x)) to_power m),(x \ y)) to_power n ) holds for E being Congruence of X for I being Ideal of X st I = Class (E,(0. X)) holds E is I-congruence of X,I
t44_bcialg_2:: for X being BCI-algebra holds IConSet X c= ConSet X
t45_bcialg_2:: for X being BCI-algebra holds ConSet X c= LConSet X
t46_bcialg_2:: for X being BCI-algebra holds ConSet X c= RConSet X
t47_bcialg_2:: for X being BCI-algebra holds ConSet X = (LConSet X) /\ (RConSet X)
t48_bcialg_2:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I for E being Congruence of X st ( for LC being L-congruence of X holds LC is I-congruence of X,I ) holds E = RI
t49_bcialg_2:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I for E being Congruence of X st ( for RC being R-congruence of X holds RC is I-congruence of X,I ) holds E = RI
t5_bcialg_2:: for X being BCI-algebra for x being Element of X for n being Element of NAT holds (x,(0. X)) to_power (n + 1) = x
t50_bcialg_2:: for X being BCI-algebra for LC being L-congruence of X holds Class (LC,(0. X)) is closed Ideal of X
t51_bcialg_2:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I holds X ./. RI is BCI-algebra
t52_bcialg_2:: for X being BCI-algebra for I being Ideal of X st I = BCK-part X holds for RI being I-congruence of X,I holds X ./. RI is p-Semisimple BCI-algebra
t6_bcialg_2:: for X being BCI-algebra for n being Element of NAT holds ((0. X),(0. X)) to_power n = 0. X
t7_bcialg_2:: for X being BCI-algebra for x, y, z being Element of X for n being Element of NAT holds ((x,y) to_power n) \ z = ((x \ z),y) to_power n
t8_bcialg_2:: for X being BCI-algebra for x, y being Element of X for n being Element of NAT holds (x,(x \ (x \ y))) to_power n = (x,y) to_power n
t9_bcialg_2:: for X being BCI-algebra for x being Element of X for n being Element of NAT holds (((0. X),x) to_power n) ` = ((0. X),(x `)) to_power n
d1_bcialg_3:: for IT being non empty BCIStr_0 holds ( IT is commutative iff for x, y being Element of IT holds x \ (x \ y) = y \ (y \ x) );
d10_bcialg_3:: for X being BCK-algebra for b2 being non empty Subset of X holds ( b2 is Commutative-Ideal of X iff ( 0. X in b2 & ( for x, y, z being Element of X st (x \ y) \ z in b2 & z in b2 holds x \ (y \ (y \ x)) in b2 ) ) );
d11_bcialg_3:: for IT being BCK-algebra holds ( IT is BCK-positive-implicative iff for x, y, z being Element of IT holds (x \ y) \ z = (x \ z) \ (y \ z) );
d12_bcialg_3:: for IT being BCK-algebra holds ( IT is BCK-implicative iff for x, y being Element of IT holds x \ (y \ x) = x );
d2_bcialg_3:: for X being BCI-algebra for a being Element of X holds ( a is being_greatest iff for x being Element of X holds x \ a = 0. X );
d3_bcialg_3:: for X being BCI-algebra for a being Element of X holds ( a is being_positive iff (0. X) \ a = 0. X );
d4_bcialg_3:: for IT being BCI-algebra holds ( IT is BCI-commutative iff for x, y being Element of IT st x \ y = 0. IT holds x = y \ (y \ x) );
d5_bcialg_3:: for IT being BCI-algebra holds ( IT is BCI-weakly-commutative iff for x, y being Element of IT holds (x \ (x \ y)) \ ((0. IT) \ (x \ y)) = y \ (y \ x) );
d6_bcialg_3:: for IT being BCK-algebra holds ( IT is bounded iff ex a being Element of IT st a is being_greatest );
d7_bcialg_3:: for IT being bounded BCK-algebra holds ( IT is involutory iff for a being Element of IT st a is being_greatest holds for x being Element of IT holds a \ (a \ x) = x );
d8_bcialg_3:: for IT being BCK-algebra for a being Element of IT holds ( a is being_Iseki iff for x being Element of IT holds ( x \ a = 0. IT & a \ x = a ) );
d9_bcialg_3:: for IT being BCK-algebra holds ( IT is Iseki_extension iff ex a being Element of IT st a is being_Iseki );
t1_bcialg_3:: for X being BCK-algebra holds ( X is commutative BCK-algebra iff for x, y being Element of X holds x \ (x \ y) <= y \ (y \ x) )
t10_bcialg_3:: for X being BCK-algebra st X is commutative BCK-algebra holds for x, y being Element of X holds ( x \ (y \ (y \ x)) = x \ y & (x \ y) \ ((x \ y) \ x) = x \ y )
t11_bcialg_3:: for X being BCK-algebra st X is commutative BCK-algebra holds for x, y, a being Element of X st x <= a holds (a \ y) \ ((a \ y) \ (a \ x)) = (a \ y) \ (x \ y) by Th8;
t12_bcialg_3:: for X being BCI-algebra st ex a being Element of X st a is being_greatest holds X is BCK-algebra
t13_bcialg_3:: for X being BCI-algebra st X is p-Semisimple holds ( X is BCI-commutative & X is BCI-weakly-commutative )
t14_bcialg_3:: for X being commutative BCK-algebra holds ( X is BCI-commutative BCI-algebra & X is BCI-weakly-commutative BCI-algebra )
t15_bcialg_3:: for X being BCK-algebra st X is BCI-weakly-commutative BCI-algebra holds X is BCI-commutative
t16_bcialg_3:: for X being BCI-algebra holds ( X is BCI-commutative iff for x, y being Element of X holds x \ (x \ y) = y \ (y \ (x \ (x \ y))) )
t17_bcialg_3:: for X being BCI-algebra holds ( X is BCI-commutative iff for x, y being Element of X holds (x \ (x \ y)) \ (y \ (y \ x)) = (0. X) \ (x \ y) )
t18_bcialg_3:: for X being BCI-algebra holds ( X is BCI-commutative iff for a being Element of AtomSet X for x, y being Element of BranchV a holds x \ (x \ y) = y \ (y \ x) )
t19_bcialg_3:: for X being non empty BCIStr_0 holds ( X is BCI-commutative BCI-algebra iff for x, y, z being Element of X holds ( ((x \ y) \ (x \ z)) \ (z \ y) = 0. X & x \ (0. X) = x & x \ (x \ y) = y \ (y \ (x \ (x \ y))) ) )
t2_bcialg_3:: for X being BCK-algebra for x, y being Element of X holds ( x \ (x \ y) <= y & x \ (x \ y) <= x )
t20_bcialg_3:: for X being BCI-algebra holds ( X is BCI-commutative iff for x, y, z being Element of X st x <= z & z \ y <= z \ x holds x <= y )
t21_bcialg_3:: for X being BCI-algebra holds ( X is BCI-commutative iff for x, y, z being Element of X st x <= y & x <= z holds x <= y \ (y \ z) )
t22_bcialg_3:: for X being bounded BCK-algebra holds ( X is involutory iff for a being Element of X st a is being_greatest holds for x, y being Element of X holds x \ y = (a \ y) \ (a \ x) )
t23_bcialg_3:: for X being bounded BCK-algebra holds ( X is involutory iff for a being Element of X st a is being_greatest holds for x, y being Element of X holds x \ (a \ y) = y \ (a \ x) )
t24_bcialg_3:: for X being bounded BCK-algebra holds ( X is involutory iff for a being Element of X st a is being_greatest holds for x, y being Element of X st x <= a \ y holds y <= a \ x )
t25_bcialg_3:: for X being BCK-algebra for IT being non empty Subset of X st IT is Commutative-Ideal of X holds for x, y being Element of X st x \ y in IT holds x \ (y \ (y \ x)) in IT
t26_bcialg_3:: for X being BCK-algebra for IT being non empty Subset of X for X being BCK-algebra st IT is Commutative-Ideal of X holds IT is Ideal of X
t27_bcialg_3:: for X being BCK-algebra for IT being non empty Subset of X st IT is Commutative-Ideal of X holds for x, y being Element of X st x \ (x \ y) in IT holds (y \ (y \ x)) \ (x \ y) in IT
t28_bcialg_3:: for X being BCK-algebra holds ( X is BCK-positive-implicative BCK-algebra iff for x, y being Element of X holds x \ y = (x \ y) \ y )
t29_bcialg_3:: for X being BCK-algebra holds ( X is BCK-positive-implicative BCK-algebra iff for x, y being Element of X holds (x \ (x \ y)) \ (y \ x) = x \ (x \ (y \ (y \ x))) )
t3_bcialg_3:: for X being BCK-algebra holds ( X is commutative BCK-algebra iff for x, y being Element of X holds x \ y = x \ (y \ (y \ x)) )
t30_bcialg_3:: for X being BCK-algebra holds ( X is BCK-positive-implicative BCK-algebra iff for x, y being Element of X holds x \ y = (x \ y) \ (x \ (x \ y)) )
t31_bcialg_3:: for X being BCK-algebra holds ( X is BCK-positive-implicative BCK-algebra iff for x, y, z being Element of X holds (x \ z) \ (y \ z) <= (x \ y) \ z )
t32_bcialg_3:: for X being BCK-algebra holds ( X is BCK-positive-implicative BCK-algebra iff for x, y being Element of X holds x \ y <= (x \ y) \ y )
t33_bcialg_3:: for X being BCK-algebra holds ( X is BCK-positive-implicative BCK-algebra iff for x, y being Element of X holds x \ (x \ (y \ (y \ x))) <= (x \ (x \ y)) \ (y \ x) )
t34_bcialg_3:: for X being BCK-algebra holds ( X is BCK-implicative BCK-algebra iff ( X is commutative BCK-algebra & X is BCK-positive-implicative BCK-algebra ) )
t35_bcialg_3:: for X being BCK-algebra holds ( X is BCK-implicative BCK-algebra iff for x, y being Element of X holds (x \ (x \ y)) \ (x \ y) = y \ (y \ x) )
t36_bcialg_3:: for X being non empty BCIStr_0 holds ( X is BCK-implicative BCK-algebra iff for x, y, z being Element of X holds ( x \ ((0. X) \ y) = x & (x \ z) \ (x \ y) = ((y \ z) \ (y \ x)) \ (x \ y) ) )
t37_bcialg_3:: for X being bounded BCK-algebra for a being Element of X st a is being_greatest holds ( X is BCK-implicative iff ( X is involutory & X is BCK-positive-implicative ) )
t38_bcialg_3:: for X being BCK-algebra holds ( X is BCK-implicative BCK-algebra iff for x, y being Element of X holds x \ (x \ (y \ x)) = 0. X )
t39_bcialg_3:: for X being BCK-algebra holds ( X is BCK-implicative BCK-algebra iff for x, y being Element of X holds (x \ (x \ y)) \ (x \ y) = y \ (y \ (x \ (x \ y))) )
t4_bcialg_3:: for X being BCK-algebra holds ( X is commutative BCK-algebra iff for x, y being Element of X holds x \ (x \ y) = y \ (y \ (x \ (x \ y))) )
t40_bcialg_3:: for X being BCK-algebra holds ( X is BCK-implicative BCK-algebra iff for x, y, z being Element of X holds (x \ z) \ (x \ y) = (y \ z) \ ((y \ x) \ z) )
t41_bcialg_3:: for X being BCK-algebra holds ( X is BCK-implicative BCK-algebra iff for x, y, z being Element of X holds x \ (x \ (y \ z)) = (y \ z) \ ((y \ z) \ (x \ z)) )
t42_bcialg_3:: for X being BCK-algebra holds ( X is BCK-implicative BCK-algebra iff for x, y being Element of X holds x \ (x \ y) = (y \ (y \ x)) \ (x \ y) )
t43_bcialg_3:: for X being commutative bounded BCK-algebra for a being Element of X st a is being_greatest holds ( X is BCK-implicative iff for x being Element of X holds (a \ x) \ ((a \ x) \ x) = 0. X )
t44_bcialg_3:: for X being commutative bounded BCK-algebra for a being Element of X st a is being_greatest holds ( X is BCK-implicative iff for x being Element of X holds x \ (a \ x) = x )
t45_bcialg_3:: for X being commutative bounded BCK-algebra for a being Element of X st a is being_greatest holds ( X is BCK-implicative iff for x being Element of X holds (a \ x) \ x = a \ x )
t46_bcialg_3:: for X being commutative bounded BCK-algebra for a being Element of X st a is being_greatest holds ( X is BCK-implicative iff for x, y being Element of X holds (a \ y) \ ((a \ y) \ x) = x \ y )
t47_bcialg_3:: for X being commutative bounded BCK-algebra for a being Element of X st a is being_greatest holds ( X is BCK-implicative iff for x, y being Element of X holds y \ (y \ x) = x \ (a \ y) )
t48_bcialg_3:: for X being commutative bounded BCK-algebra for a being Element of X st a is being_greatest holds ( X is BCK-implicative iff for x, y, z being Element of X holds (x \ (y \ z)) \ (x \ y) <= x \ (a \ z) )
t5_bcialg_3:: for X being BCK-algebra holds ( X is commutative BCK-algebra iff for x, y being Element of X st x <= y holds x = y \ (y \ x) )
t6_bcialg_3:: for X being non empty BCIStr_0 holds ( X is commutative BCK-algebra iff for x, y, z being Element of X holds ( x \ ((0. X) \ y) = x & (x \ z) \ (x \ y) = (y \ z) \ (y \ x) ) )
t7_bcialg_3:: for X being BCK-algebra st X is commutative BCK-algebra holds for x, y being Element of X st x \ y = x holds y \ x = y
t8_bcialg_3:: for X being BCK-algebra st X is commutative BCK-algebra holds for x, y, a being Element of X st y <= a holds (a \ x) \ (a \ y) = y \ x
t9_bcialg_3:: for X being BCK-algebra st X is commutative BCK-algebra holds for x, y being Element of X holds ( x \ y = x iff y \ (y \ x) = 0. X )
d1_bcialg_4:: for A being BCIStr_1 for x, y being Element of A holds x * y = the ExternalDiff of A . (x,y);
d10_bcialg_4:: for X being BCI-algebra for a being Element of X holds Initial_section a = { t where t is Element of X : t <= a } ;
d11_bcialg_4:: for IT being BCK-Algebra_with_Condition(S) holds ( IT is positive-implicative iff for x, y being Element of IT holds (x \ y) \ y = x \ y );
d12_bcialg_4:: for IT being non empty BCIStr_1 holds ( IT is being_SB-1 iff for x being Element of IT holds x * x = x );
d13_bcialg_4:: for IT being non empty BCIStr_1 holds ( IT is being_SB-2 iff for x, y being Element of IT holds x * y = y * x );
d14_bcialg_4:: for IT being non empty BCIStr_1 holds ( IT is being_SB-4 iff for x, y being Element of IT holds (x \ y) * y = x * y );
d15_bcialg_4:: for IT being BCK-Algebra_with_Condition(S) holds ( IT is implicative iff for x, y being Element of IT holds x \ (y \ x) = x );
d2_bcialg_4:: for IT being non empty BCIStr_1 holds ( IT is with_condition_S iff for x, y, z being Element of IT holds (x \ y) \ z = x \ (y * z) );
d3_bcialg_4:: BCI_S-EXAMPLE = BCIStr_1(# 1,op2,op2,op0 #);
d4_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y being Element of X holds Condition_S (x,y) = { t where t is Element of X : t \ x <= y } ;
d5_bcialg_4:: for X being p-Semisimple BCI-algebra for b2 being strict AbGroup holds ( b2 = Adjoint_pGroup X iff ( the carrier of b2 = the carrier of X & ( for x, y being Element of X holds the addF of b2 . (x,y) = x \ ((0. X) \ y) ) & 0. b2 = 0. X ) );
d6_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for b2 being Function of [: the carrier of X,NAT:], the carrier of X holds ( b2 = power X iff for h being Element of X holds ( b2 . (h,0) = 0. X & ( for n being Element of NAT holds b2 . (h,(n + 1)) = (b2 . (h,n)) * h ) ) );
d7_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x being Element of X for n being Element of NAT holds x |^ n = (power X) . (x,n);
d8_bcialg_4:: for X being non empty BCIStr_1 for F being FinSequence of the carrier of X holds Product_S F = the ExternalDiff of X "**" F;
d9_bcialg_4:: for IT being BCK-Algebra_with_Condition(S) holds ( IT is commutative iff for x, y being Element of IT holds x \ (x \ y) = y \ (y \ x) );
t1_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, u, v being Element of X st u in Condition_S (x,y) & v <= u holds v in Condition_S (x,y)
t10_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, z being Element of X holds (x * y) * z = x * (y * z)
t11_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, z being Element of X holds (x * y) * z = (x * z) * y
t12_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, z being Element of X holds (x \ y) \ z = x \ (y * z)
t13_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y being Element of X holds y <= x * (y \ x)
t14_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, z being Element of X holds (x * z) \ (y * z) <= x \ y
t15_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, z being Element of X holds ( x \ y <= z iff x <= y * z )
t16_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, z being Element of X holds x \ y <= (x \ z) * (z \ y)
t17_bcialg_4:: for X being BCI-Algebra_with_Condition(S) holds 0. X is_a_unity_wrt the ExternalDiff of X
t18_bcialg_4:: for X being BCI-Algebra_with_Condition(S) holds the_unity_wrt the ExternalDiff of X = 0. X
t19_bcialg_4:: for X being BCI-Algebra_with_Condition(S) holds the ExternalDiff of X is having_a_unity
t2_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y being Element of X ex a being Element of Condition_S (x,y) st for z being Element of Condition_S (x,y) holds z <= a
t20_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x being Element of X holds x |^ 0 = 0. X by Def6;
t21_bcialg_4:: for n being Element of NAT for X being BCI-Algebra_with_Condition(S) for x being Element of X holds x |^ (n + 1) = (x |^ n) * x by Def6;
t22_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x being Element of X holds x |^ 1 = x
t23_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x being Element of X holds x |^ 2 = x * x
t24_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x being Element of X holds x |^ 3 = (x * x) * x
t25_bcialg_4:: for X being BCI-Algebra_with_Condition(S) holds (0. X) |^ 2 = 0. X
t26_bcialg_4:: for n being Element of NAT for X being BCI-Algebra_with_Condition(S) holds (0. X) |^ n = 0. X
t27_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, a being Element of X holds ((x \ a) \ a) \ a = x \ (a |^ 3)
t28_bcialg_4:: for n being Element of NAT for X being BCI-Algebra_with_Condition(S) for x, a being Element of X holds (x,a) to_power n = x \ (a |^ n)
t29_bcialg_4:: for X being non empty BCIStr_1 for d being Element of X holds the ExternalDiff of X "**" <*d*> = d
t3_bcialg_4:: for X being non empty BCIStr_1 holds ( ( X is BCI-algebra & ( for x, y being Element of X holds ( (x * y) \ x <= y & ( for t being Element of X st t \ x <= y holds t <= x * y ) ) ) ) iff X is BCI-Algebra_with_Condition(S) ) by Lm2, Lm3;
t30_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for F1, F2 being FinSequence of the carrier of X holds Product_S (F1 ^ F2) = (Product_S F1) * (Product_S F2) by Th19, FINSOP_1:5;
t31_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for F being FinSequence of the carrier of X for a being Element of X holds Product_S (F ^ <*a*>) = (Product_S F) * a by Th19, FINSOP_1:4;
t32_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for F being FinSequence of the carrier of X for a being Element of X holds Product_S (<*a*> ^ F) = a * (Product_S F) by Th19, FINSOP_1:6;
t33_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for a1, a2 being Element of X holds Product_S <*a1,a2*> = a1 * a2
t34_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for a1, a2, a3 being Element of X holds Product_S <*a1,a2,a3*> = (a1 * a2) * a3
t35_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, a1, a2 being Element of X holds (x \ a1) \ a2 = x \ (Product_S <*a1,a2*>)
t36_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, a1, a2, a3 being Element of X holds ((x \ a1) \ a2) \ a3 = x \ (Product_S <*a1,a2,a3*>)
t37_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for a, b being Element of AtomSet X for ma being Element of X st ( for x being Element of BranchV a holds x <= ma ) holds ex mb being Element of X st for y being Element of BranchV b holds y <= mb
t38_bcialg_4:: for X being BCK-Algebra_with_Condition(S) for x, y being Element of X holds ( x <= x * y & y <= x * y )
t39_bcialg_4:: for X being BCK-Algebra_with_Condition(S) for x, y, z being Element of X holds ((x * y) \ (y * z)) \ (z * x) = 0. X
t4_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y being Element of X ex a being Element of Condition_S (x,y) st for z being Element of Condition_S (x,y) holds z <= a
t40_bcialg_4:: for X being BCK-Algebra_with_Condition(S) for x, y being Element of X holds (x \ y) * (y \ x) <= x * y
t41_bcialg_4:: for X being BCK-Algebra_with_Condition(S) for x being Element of X holds (x \ (0. X)) * ((0. X) \ x) = x
t42_bcialg_4:: for X being non empty BCIStr_1 holds ( X is commutative BCK-Algebra_with_Condition(S) iff for x, y, z being Element of X holds ( x \ ((0. X) \ y) = x & (x \ z) \ (x \ y) = (y \ z) \ (y \ x) & (x \ y) \ z = x \ (y * z) ) )
t43_bcialg_4:: for X being commutative BCK-Algebra_with_Condition(S) for a being Element of X st a is greatest holds for x, y being Element of X holds x * y = a \ ((a \ x) \ y)
t44_bcialg_4:: for X being commutative BCK-Algebra_with_Condition(S) for a, b, c being Element of X st Condition_S (a,b) c= Initial_section c holds for x being Element of Condition_S (a,b) holds x <= c \ ((c \ a) \ b)
t45_bcialg_4:: for X being BCK-Algebra_with_Condition(S) holds ( X is positive-implicative iff for x being Element of X holds x * x = x )
t46_bcialg_4:: for X being BCK-Algebra_with_Condition(S) holds ( X is positive-implicative iff for x, y being Element of X st x <= y holds x * y = y )
t47_bcialg_4:: for X being BCK-Algebra_with_Condition(S) holds ( X is positive-implicative iff for x, y, z being Element of X holds (x * y) \ z = (x \ z) * (y \ z) )
t48_bcialg_4:: for X being BCK-Algebra_with_Condition(S) holds ( X is positive-implicative iff for x, y being Element of X holds x * y = x * (y \ x) )
t49_bcialg_4:: for X being positive-implicative BCK-Algebra_with_Condition(S) for x, y being Element of X holds x = (x \ y) * (x \ (x \ y))
t5_bcialg_4:: for X being BCI-algebra holds ( X is p-Semisimple iff for x, y being Element of X st x \ y = 0. X holds x = y )
t50_bcialg_4:: for X being non empty BCIStr_1 holds ( X is positive-implicative BCK-Algebra_with_Condition(S) iff X is semi-Brouwerian-algebra )
t51_bcialg_4:: for X being BCK-Algebra_with_Condition(S) holds ( X is implicative iff ( X is commutative & X is positive-implicative ) )
t52_bcialg_4:: for X being BCK-Algebra_with_Condition(S) holds ( X is implicative iff for x, y, z being Element of X holds x \ (y \ z) = ((x \ y) \ z) * (z \ (z \ x)) )
t6_bcialg_4:: for X being BCI-Algebra_with_Condition(S) st X is p-Semisimple holds for x, y being Element of X holds x * y = x \ ((0. X) \ y)
t7_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y being Element of X holds x * y = y * x
t8_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x, y, z being Element of X st x <= y holds ( x * z <= y * z & z * x <= z * y )
t9_bcialg_4:: for X being BCI-Algebra_with_Condition(S) for x being Element of X holds ( (0. X) * x = x & x * (0. X) = x )
d1_bcialg_5:: for X being BCI-algebra for x, y being Element of X for m, n being Element of NAT holds Polynom (m,n,x,y) = (((x,(x \ y)) to_power (m + 1)),(y \ x)) to_power n;
d2_bcialg_5:: for X being BCI-algebra holds ( X is quasi-commutative iff ex i, j, m, n being Element of NAT st for x, y being Element of X holds Polynom (i,j,x,y) = Polynom (m,n,y,x) );
d3_bcialg_5:: for i, j, m, n being Element of NAT for b5 being BCI-algebra holds ( b5 is BCI-algebra of i,j,m,n iff for x, y being Element of b5 holds Polynom (i,j,x,y) = Polynom (m,n,y,x) );
d4_bcialg_5:: for i, j, m, n being Element of NAT holds min (i,j,m,n) = min ((min (i,j)),(min (m,n)));
d5_bcialg_5:: for i, j, m, n being Element of NAT holds max (i,j,m,n) = max ((max (i,j)),(max (m,n)));
t1_bcialg_5:: for X being BCI-algebra for x, y, z being Element of X st x <= y & y <= z holds x <= z
t10_bcialg_5:: for X being BCI-algebra for x, y being Element of X for m, n being Element of NAT holds Polynom (m,(n + 1),x,y) = (Polynom (m,n,x,y)) \ (y \ x)
t11_bcialg_5:: for X being BCI-algebra for y, x being Element of X for n being Element of NAT holds Polynom ((n + 1),(n + 1),y,x) <= Polynom (n,(n + 1),x,y)
t12_bcialg_5:: for X being BCI-algebra for x, y being Element of X for n being Element of NAT holds Polynom (n,(n + 1),x,y) <= Polynom (n,n,y,x)
t13_bcialg_5:: for X being BCI-algebra for i, j, m, n being Element of NAT holds ( X is BCI-algebra of i,j,m,n iff X is BCI-algebra of m,n,i,j )
t14_bcialg_5:: for i, j, m, n being Element of NAT for X being BCI-algebra of i,j,m,n for k being Element of NAT holds X is BCI-algebra of i + k,j,m,n + k
t15_bcialg_5:: for i, j, m, n being Element of NAT for X being BCI-algebra of i,j,m,n for k being Element of NAT holds X is BCI-algebra of i,j + k,m + k,n
t16_bcialg_5:: for X being BCI-algebra for i, j, m, n being Element of NAT holds ( X is BCK-algebra of i,j,m,n iff X is BCK-algebra of m,n,i,j )
t17_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n for k being Element of NAT holds X is BCK-algebra of i + k,j,m,n + k
t18_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n for k being Element of NAT holds X is BCK-algebra of i,j + k,m + k,n
t19_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n for x, y being Element of X holds (x,y) to_power (i + 1) = (x,y) to_power (n + 1)
t2_bcialg_5:: for X being BCI-algebra for x, y being Element of X st x <= y & y <= x holds x = y
t20_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n for x, y being Element of X holds (x,y) to_power (j + 1) = (x,y) to_power (m + 1)
t21_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n holds X is BCK-algebra of i,j,j,n
t22_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n holds X is BCK-algebra of n,j,m,n
t23_bcialg_5:: for i, j, m, n being Element of NAT holds ( min (i,j,m,n) = i or min (i,j,m,n) = j or min (i,j,m,n) = m or min (i,j,m,n) = n )
t24_bcialg_5:: for i, j, m, n being Element of NAT holds ( max (i,j,m,n) = i or max (i,j,m,n) = j or max (i,j,m,n) = m or max (i,j,m,n) = n )
t25_bcialg_5:: for i, j, m, n being Element of NAT st i = min (i,j,m,n) holds ( i <= j & i <= m & i <= n )
t26_bcialg_5:: for i, j, m, n being Element of NAT holds ( max (i,j,m,n) >= i & max (i,j,m,n) >= j & max (i,j,m,n) >= m & max (i,j,m,n) >= n )
t27_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i = min (i,j,m,n) & i = j holds X is BCK-algebra of i,i,i,i
t28_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i = min (i,j,m,n) & i < j & i < n holds X is BCK-algebra of i,i + 1,i,i + 1
t29_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i = min (i,j,m,n) & i = n & i = m holds X is BCK-algebra of i,i,i,i
t3_bcialg_5:: for n being Element of NAT for X being BCK-algebra for x, y being Element of X holds ( x \ y <= x & (x,y) to_power (n + 1) <= (x,y) to_power n )
t30_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i = n & m < j holds X is BCK-algebra of i,m + 1,m,i
t31_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i = n holds X is BCK-algebra of i,j,j,i
t32_bcialg_5:: for i, j, m, n, l, k being Element of NAT for X being BCK-algebra of i,j,m,n st l >= j & k >= n holds X is BCK-algebra of k,l,l,k
t33_bcialg_5:: for i, j, m, n, k being Element of NAT for X being BCK-algebra of i,j,m,n st k >= max (i,j,m,n) holds X is BCK-algebra of k,k,k,k
t34_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i <= m & j <= n holds X is BCK-algebra of i,j,i,j
t35_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i <= m & i < n holds X is BCK-algebra of i,j,i,i + 1
t36_bcialg_5:: for X being BCI-algebra for i, j, k being Element of NAT st X is BCI-algebra of i,j,j + k,i + k holds X is BCK-algebra
t37_bcialg_5:: for X being BCI-algebra holds ( X is BCI-algebra of 0 , 0 , 0 , 0 iff X is BCK-algebra of 0 , 0 , 0 , 0 )
t38_bcialg_5:: for X being BCI-algebra holds ( X is commutative BCK-algebra iff X is BCI-algebra of 0 , 0 , 0 , 0 )
t39_bcialg_5:: for X being BCI-algebra for B, P being non empty Subset of X for X being BCI-algebra st B = BCK-part X & P = p-Semisimple-part X holds B /\ P = {(0. X)}
t4_bcialg_5:: for n being Element of NAT for X being BCK-algebra for x being Element of X holds ((0. X),x) to_power n = 0. X
t40_bcialg_5:: for X being BCI-algebra for P being non empty Subset of X for X being BCI-algebra st P = p-Semisimple-part X holds ( X is BCK-algebra iff P = {(0. X)} )
t41_bcialg_5:: for X being BCI-algebra for B being non empty Subset of X for X being BCI-algebra st B = BCK-part X holds ( X is p-Semisimple BCI-algebra iff B = {(0. X)} )
t42_bcialg_5:: for X being BCI-algebra st X is p-Semisimple BCI-algebra holds X is BCI-algebra of 0 ,1, 0 , 0
t43_bcialg_5:: for X being BCI-algebra for n, j, m being Element of NAT st X is p-Semisimple BCI-algebra holds X is BCI-algebra of n + j,n,m,(m + j) + 1
t44_bcialg_5:: for X being BCI-algebra st X is associative BCI-algebra holds ( X is BCI-algebra of 0 ,1, 0 , 0 & X is BCI-algebra of 1, 0 , 0 , 0 )
t45_bcialg_5:: for X being BCI-algebra st X is weakly-positive-implicative BCI-algebra holds X is BCI-algebra of 0 ,1,1,1
t46_bcialg_5:: for X being BCI-algebra st X is positive-implicative BCI-algebra holds X is BCI-algebra of 0 ,1,1,1
t47_bcialg_5:: for X being BCI-algebra st X is implicative BCI-algebra holds X is BCI-algebra of 0 ,1, 0 , 0
t48_bcialg_5:: for X being BCI-algebra st X is alternative BCI-algebra holds X is BCI-algebra of 0 ,1, 0 , 0
t49_bcialg_5:: for X being BCI-algebra holds ( X is BCK-positive-implicative BCK-algebra iff X is BCK-algebra of 0 ,1, 0 ,1 )
t5_bcialg_5:: for m, n being Element of NAT for X being BCK-algebra for x, y being Element of X st m >= n holds (x,y) to_power m <= (x,y) to_power n
t50_bcialg_5:: for X being BCI-algebra holds ( X is BCK-implicative BCK-algebra iff X is BCK-algebra of 1, 0 , 0 , 0 )
t51_bcialg_5:: for X being BCI-algebra holds ( X is BCK-algebra of 1, 0 , 0 , 0 iff ( X is BCK-algebra of 0 , 0 , 0 , 0 & X is BCK-algebra of 0 ,1, 0 ,1 ) )
t52_bcialg_5:: for X being quasi-commutative BCK-algebra holds ( X is BCK-algebra of 0 ,1, 0 ,1 iff for x, y being Element of X holds x \ y = (x \ y) \ y )
t53_bcialg_5:: for n being Element of NAT for X being quasi-commutative BCK-algebra holds ( X is BCK-algebra of n,n + 1,n,n + 1 iff for x, y being Element of X holds (x,y) to_power (n + 1) = (x,y) to_power (n + 2) )
t54_bcialg_5:: for X being BCI-algebra st X is BCI-algebra of 0 ,1, 0 , 0 holds X is BCI-commutative BCI-algebra
t55_bcialg_5:: for X being BCI-algebra for n, m being Element of NAT st X is BCI-algebra of n, 0 ,m,m holds X is BCI-commutative BCI-algebra
t56_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st j = 0 & m > 0 holds X is BCK-algebra of 0 , 0 , 0 , 0
t57_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st m = 0 & j > 0 holds X is BCK-algebra of 0 ,1, 0 ,1
t58_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st n = 0 & i <> 0 holds X is BCK-algebra of 0 , 0 , 0 , 0
t59_bcialg_5:: for i, j, m, n being Element of NAT for X being BCK-algebra of i,j,m,n st i = 0 & n <> 0 holds X is BCK-algebra of 0 ,1, 0 ,1
t6_bcialg_5:: for m, n being Element of NAT for X being BCK-algebra for x, y being Element of X st m > n & (x,y) to_power n = (x,y) to_power m holds for k being Element of NAT st k >= n holds (x,y) to_power n = (x,y) to_power k
t7_bcialg_5:: for X being BCI-algebra for x, y being Element of X holds Polynom (0,0,x,y) = x \ (x \ y)
t8_bcialg_5:: for X being BCI-algebra for x, y being Element of X for m, n being Element of NAT holds Polynom (m,n,x,y) = ((((Polynom (0,0,x,y)),(x \ y)) to_power m),(y \ x)) to_power n
t9_bcialg_5:: for X being BCI-algebra for x, y being Element of X for m, n being Element of NAT holds Polynom ((m + 1),n,x,y) = (Polynom (m,n,x,y)) \ (x \ y)
d1_bcialg_6:: for G being non empty BCIStr_0 for b2 being Function of [: the carrier of G,NAT:], the carrier of G holds ( b2 = BCI-power G iff for x being Element of G holds ( b2 . (x,0) = 0. G & ( for n being Element of NAT holds b2 . (x,(n + 1)) = x \ ((b2 . (x,n)) `) ) ) );
d10_bcialg_6:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I for b4 being BCI-homomorphism of X,(X ./. RI) holds ( b4 = nat_hom RI iff for x being Element of X holds b4 . x = Class (RI,x) );
d11_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K holds Union (G,RK) = union { (Class (RK,a)) where a is Element of G : Class (RK,a) in the carrier of (X ./. RK) } ;
d12_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K for b5 being BinOp of (Union (G,RK)) holds ( b5 = HKOp (G,RK) iff for w1, w2 being Element of Union (G,RK) for x, y being Element of X st w1 = x & w2 = y holds b5 . (w1,w2) = x \ y );
d13_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K holds zeroHK (G,RK) = 0. X;
d14_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K holds HK (G,RK) = BCIStr_0(# (Union (G,RK)),(HKOp (G,RK)),(zeroHK (G,RK)) #);
d15_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K for w1, w2 being Element of Union (G,RK) holds w1 \ w2 = (HKOp (G,RK)) . (w1,w2);
d2_bcialg_6:: for X being BCI-algebra for i being Integer for x being Element of X holds ( ( 0 <= i implies x |^ i = (BCI-power X) . (x,(abs i)) ) & ( not 0 <= i implies x |^ i = (BCI-power X) . ((x `),(abs i)) ) );
d3_bcialg_6:: for X being BCI-algebra for n being Nat for x being Element of X holds x |^ n = (BCI-power X) . (x,n);
d4_bcialg_6:: for X being BCI-algebra for x being Element of X holds ( x is finite-period iff ex n being Element of NAT st ( n <> 0 & x |^ n in BCK-part X ) );
d5_bcialg_6:: for X being BCI-algebra for x being Element of X st x is finite-period holds for b3 being Element of NAT holds ( b3 = ord x iff ( x |^ b3 in BCK-part X & b3 <> 0 & ( for m being Element of NAT st x |^ m in BCK-part X & m <> 0 holds b3 <= m ) ) );
d6_bcialg_6:: for X, X9 being non empty BCIStr_0 for f being Function of X,X9 holds ( f is multiplicative iff for a, b being Element of X holds f . (a \ b) = (f . a) \ (f . b) );
d7_bcialg_6:: for X, X9 being BCI-algebra for f being BCI-homomorphism of X,X9 holds ( f is isotonic iff for x, y being Element of X st x <= y holds f . x <= f . y );
d8_bcialg_6:: for X, X9 being BCI-algebra for f being BCI-homomorphism of X,X9 holds Ker f = { x where x is Element of X : f . x = 0. X9 } ;
d9_bcialg_6:: for X, X9 being BCI-algebra holds ( X,X9 are_isomorphic iff ex f being BCI-homomorphism of X,X9 st f is bijective );
t1_bcialg_6:: for X being BCI-algebra for x being Element of X for a, b being Element of AtomSet X holds a \ (x \ b) = b \ (x \ a)
t10_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for n being Nat holds (a `) |^ n = a |^ (- n)
t11_bcialg_6:: for X being BCI-algebra for x being Element of X for n being Nat st x in BCK-part X & n >= 1 holds x |^ n = x
t12_bcialg_6:: for X being BCI-algebra for x being Element of X for n being Nat st x in BCK-part X holds x |^ (- n) = 0. X
t13_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for i being Integer holds a |^ i in AtomSet X
t14_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for n being Nat holds (a |^ (n + 1)) ` = ((a |^ n) `) \ a
t15_bcialg_6:: for X being BCI-algebra for a, b being Element of AtomSet X for n being Nat holds (a \ b) |^ n = (a |^ n) \ (b |^ n)
t16_bcialg_6:: for X being BCI-algebra for a, b being Element of AtomSet X for n being Nat holds (a \ b) |^ (- n) = (a |^ (- n)) \ (b |^ (- n))
t17_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for n being Nat holds (a `) |^ n = (a |^ n) `
t18_bcialg_6:: for X being BCI-algebra for x being Element of X for n being Nat holds (x `) |^ n = (x |^ n) `
t19_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for n being Nat holds (a `) |^ (- n) = (a |^ (- n)) `
t2_bcialg_6:: for X being BCI-algebra for x being Element of X for n being Nat holds x |^ (n + 1) = x \ ((x |^ n) `)
t20_bcialg_6:: for X being BCI-algebra for x being Element of X for a being Element of AtomSet X for n being Nat st a = ((x `) `) |^ n holds x |^ n in BranchV a
t21_bcialg_6:: for X being BCI-algebra for x being Element of X for n being Nat holds (x |^ n) ` = (((x `) `) |^ n) `
t22_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for i, j being Integer holds (a |^ i) \ (a |^ j) = a |^ (i - j)
t23_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for i, j being Integer holds (a |^ i) |^ j = a |^ (i * j)
t24_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for i, j being Integer holds a |^ (i + j) = (a |^ i) \ ((a |^ j) `)
t25_bcialg_6:: for X being BCI-algebra for x being Element of X st x is finite-period holds (x `) ` is finite-period
t26_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X for n being Nat st a is finite-period & ord a = n holds a |^ n = 0. X
t27_bcialg_6:: for X being BCI-algebra holds ( X is BCK-algebra iff for x being Element of X holds ( x is finite-period & ord x = 1 ) )
t28_bcialg_6:: for X being BCI-algebra for x being Element of X for a being Element of AtomSet X st x is finite-period & a is finite-period & x in BranchV a holds ord x = ord a
t29_bcialg_6:: for X being BCI-algebra for x being Element of X for n, m being Nat st x is finite-period & ord x = n holds ( x |^ m in BCK-part X iff n divides m )
t3_bcialg_6:: for X being BCI-algebra for x being Element of X holds x |^ 0 = 0. X by Def1;
t30_bcialg_6:: for X being BCI-algebra for x being Element of X for m, n being Nat st x is finite-period & x |^ m is finite-period & ord x = n & m > 0 holds ord (x |^ m) = n div (m gcd n)
t31_bcialg_6:: for X being BCI-algebra for x being Element of X st x is finite-period & x ` is finite-period holds ord x = ord (x `)
t32_bcialg_6:: for X being BCI-algebra for x, y being Element of X for a being Element of AtomSet X st x \ y is finite-period & x in BranchV a & y in BranchV a holds ord (x \ y) = 1
t33_bcialg_6:: for X being BCI-algebra for x, y being Element of X for a, b being Element of AtomSet X st a \ b is finite-period & x is finite-period & y is finite-period & a is finite-period & b is finite-period & x in BranchV a & y in BranchV b holds ord (a \ b) divides (ord x) lcm (ord y)
t34_bcialg_6:: for X being BCI-algebra for Y being SubAlgebra of X for x, y being Element of X for x9, y9 being Element of Y st x = x9 & y = y9 holds x \ y = x9 \ y9
t35_bcialg_6:: for X, X9 being BCI-algebra for f being BCI-homomorphism of X,X9 holds f . (0. X) = 0. X9
t36_bcialg_6:: for X, X9 being BCI-algebra for x, y being Element of X for f being BCI-homomorphism of X,X9 st x <= y holds f . x <= f . y
t37_bcialg_6:: for X9, X being BCI-algebra for f being BCI-homomorphism of X,X9 holds ( f is one-to-one iff Ker f = {(0. X)} )
t38_bcialg_6:: for X9, X being BCI-algebra for f being BCI-homomorphism of X,X9 for g being BCI-homomorphism of X9,X st f is bijective & g = f " holds g is bijective
t39_bcialg_6:: for X9, X, Y being BCI-algebra for f being BCI-homomorphism of X,X9 for h being BCI-homomorphism of X9,Y holds h * f is BCI-homomorphism of X,Y
t4_bcialg_6:: for X being BCI-algebra for x being Element of X holds x |^ 1 = x
t40_bcialg_6:: for X9, X being BCI-algebra for f being BCI-homomorphism of X,X9 for Z being SubAlgebra of X9 st the carrier of Z = rng f holds f is BCI-homomorphism of X,Z
t41_bcialg_6:: for X9, X being BCI-algebra for f being BCI-homomorphism of X,X9 holds Ker f is closed Ideal of X
t42_bcialg_6:: for X9, X being BCI-algebra for f being BCI-homomorphism of X,X9 st f is onto holds for c being Element of X9 ex x being Element of X st c = f . x
t43_bcialg_6:: for X9, X being BCI-algebra for f being BCI-homomorphism of X,X9 for a being Element of X st a is minimal holds f . a is minimal
t44_bcialg_6:: for X, X9 being BCI-algebra for f being BCI-homomorphism of X,X9 for a being Element of AtomSet X for b being Element of AtomSet X9 st b = f . a holds f .: (BranchV a) c= BranchV b
t45_bcialg_6:: for X9, X being BCI-algebra for A9 being non empty Subset of X9 for f being BCI-homomorphism of X,X9 st A9 is Ideal of X9 holds f " A9 is Ideal of X
t46_bcialg_6:: for X9, X being BCI-algebra for A9 being non empty Subset of X9 for f being BCI-homomorphism of X,X9 st A9 is closed Ideal of X9 holds f " A9 is closed Ideal of X
t47_bcialg_6:: for X, X9 being BCI-algebra for I being Ideal of X for f being BCI-homomorphism of X,X9 st f is onto holds f .: I is Ideal of X9
t48_bcialg_6:: for X, X9 being BCI-algebra for CI being closed Ideal of X for f being BCI-homomorphism of X,X9 st f is onto holds f .: CI is closed Ideal of X9
t49_bcialg_6:: for X being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I holds nat_hom RI is onto
t5_bcialg_6:: for X being BCI-algebra for x being Element of X holds x |^ (- 1) = x `
t50_bcialg_6:: for X, X9 being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I for f being BCI-homomorphism of X,X9 st I = Ker f holds ex h being BCI-homomorphism of (X ./. RI),X9 st ( f = h * (nat_hom RI) & h is one-to-one )
t51_bcialg_6:: for X, X9 being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I for f being BCI-homomorphism of X,X9 st I = Ker f holds ex h being BCI-homomorphism of (X ./. RI),X9 st ( f = h * (nat_hom RI) & h is one-to-one ) by Th50;
t52_bcialg_6:: for X being BCI-algebra for K being closed Ideal of X for RK being I-congruence of X,K holds Ker (nat_hom RK) = K
t53_bcialg_6:: for X9, X being BCI-algebra for H9 being SubAlgebra of X9 for I being Ideal of X for RI being I-congruence of X,I for f being BCI-homomorphism of X,X9 st I = Ker f & the carrier of H9 = rng f holds X ./. RI,H9 are_isomorphic
t54_bcialg_6:: for X, X9 being BCI-algebra for I being Ideal of X for RI being I-congruence of X,I for f being BCI-homomorphism of X,X9 st I = Ker f & f is onto holds X ./. RI,X9 are_isomorphic
t55_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K holds HK (G,RK) is BCI-algebra
t56_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K holds HK (G,RK) is SubAlgebra of X
t57_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X holds the carrier of G /\ K is closed Ideal of G
t58_bcialg_6:: for X being BCI-algebra for G being SubAlgebra of X for K being closed Ideal of X for RK being I-congruence of X,K for K1 being Ideal of HK (G,RK) for RK1 being I-congruence of HK (G,RK),K1 for I being Ideal of G for RI being I-congruence of G,I st RK1 = RK & I = the carrier of G /\ K holds G ./. RI,(HK (G,RK)) ./. RK1 are_isomorphic
t6_bcialg_6:: for X being BCI-algebra for x being Element of X holds x |^ 2 = x \ (x `)
t7_bcialg_6:: for X being BCI-algebra for n being Nat holds (0. X) |^ n = 0. X
t8_bcialg_6:: for X being BCI-algebra for a being Element of AtomSet X holds (a |^ (- 1)) |^ (- 1) = a
t9_bcialg_6:: for X being BCI-algebra for x being Element of X for n being Nat holds x |^ (- n) = ((x `) `) |^ (- n)
d1_bciideal:: for X being BCI-algebra for a being Element of X holds initial_section a = { x where x is Element of X : x <= a } ;
d2_bciideal:: for X being BCI-algebra for I being Ideal of X holds ( I is positive iff for x being Element of I holds x is positive );
d3_bciideal:: for X being BCI-algebra for IT being Ideal of X holds ( IT is associative iff ( 0. X in IT & ( for x, y, z being Element of X st x \ (y \ z) in IT & y \ z in IT holds x in IT ) ) );
d4_bciideal:: for X being BCI-algebra for b2 being non empty Subset of X holds ( b2 is associative-ideal of X iff ( 0. X in b2 & ( for x, y, z being Element of X st (x \ y) \ z in b2 & y \ z in b2 holds x in b2 ) ) );
d5_bciideal:: for X being BCI-algebra for b2 being non empty Subset of X holds ( b2 is p-ideal of X iff ( 0. X in b2 & ( for x, y, z being Element of X st (x \ z) \ (y \ z) in b2 & y in b2 holds x in b2 ) ) );
d6_bciideal:: for X being BCK-algebra for IT being Ideal of X holds ( IT is commutative iff for x, y, z being Element of X st (x \ y) \ z in IT & z in IT holds x \ (y \ (y \ x)) in IT );
d7_bciideal:: for X being BCK-algebra for b2 being non empty Subset of X holds ( b2 is implicative-ideal of X iff ( 0. X in b2 & ( for x, y, z being Element of X st (x \ (y \ x)) \ z in b2 & z in b2 holds x in b2 ) ) );
d8_bciideal:: for X being BCK-algebra for b2 being non empty Subset of X holds ( b2 is positive-implicative-ideal of X iff ( 0. X in b2 & ( for x, y, z being Element of X st (x \ y) \ z in b2 & y \ z in b2 holds x \ z in b2 ) ) );
t1_bciideal:: for X being BCI-algebra for x, y, z, u being Element of X st x <= y holds u \ (z \ x) <= u \ (z \ y)
t10_bciideal:: for X being BCI-algebra st AtomSet X is Ideal of X holds AtomSet X is closed Ideal of X
t11_bciideal:: for X being BCK-algebra for A, I being Ideal of X holds ( A /\ I = {(0. X)} iff for x being Element of A for y being Element of I holds x \ y = x )
t12_bciideal:: for X being associative BCI-algebra for A being Ideal of X holds A is closed
t13_bciideal:: for X being BCI-algebra for A being Ideal of X st X is quasi-associative holds A is closed
t14_bciideal:: for X being BCI-algebra for X1 being non empty Subset of X st X1 is associative-ideal of X holds X1 is Ideal of X
t15_bciideal:: for X being BCI-algebra for I being Ideal of X holds ( I is associative-ideal of X iff for x, y, z being Element of X st (x \ y) \ z in I holds x \ (y \ z) in I )
t16_bciideal:: for X being BCI-algebra for I being Ideal of X st I is associative-ideal of X holds for x being Element of X holds x \ ((0. X) \ x) in I
t17_bciideal:: for X being BCI-algebra for I being Ideal of X st ( for x being Element of X holds x \ ((0. X) \ x) in I ) holds I is closed Ideal of X
t18_bciideal:: for X being BCI-algebra for X1 being non empty Subset of X st X1 is p-ideal of X holds X1 is Ideal of X
t19_bciideal:: for X being BCI-algebra for I being Ideal of X st I is p-ideal of X holds BCK-part X c= I
t2_bciideal:: for X being BCI-algebra for x, y, z, u being Element of X holds (x \ (y \ z)) \ (x \ (y \ u)) <= z \ u
t20_bciideal:: for X being BCI-algebra holds BCK-part X is p-ideal of X
t21_bciideal:: for X being BCI-algebra for I being Ideal of X holds ( I is p-ideal of X iff for x, y being Element of X st x in I & x <= y holds y in I )
t22_bciideal:: for X being BCI-algebra for I being Ideal of X holds ( I is p-ideal of X iff for x, y, z being Element of X st (x \ z) \ (y \ z) in I holds x \ y in I )
t23_bciideal:: for X being BCK-algebra holds BCK-part X is commutative Ideal of X
t24_bciideal:: for X being BCK-algebra st X is p-Semisimple BCI-algebra holds {(0. X)} is commutative Ideal of X
t25_bciideal:: for X being BCK-algebra holds BCK-part X = the carrier of X
t26_bciideal:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds (x \ y) \ y = x \ y ) holds the carrier of X = BCK-part X
t27_bciideal:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds x \ (y \ x) = x ) holds the carrier of X = BCK-part X
t28_bciideal:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds x \ (x \ y) = y \ (y \ x) ) holds the carrier of X = BCK-part X
t29_bciideal:: for X being BCI-algebra st ( for X being BCI-algebra for x, y, z being Element of X holds (x \ y) \ y = (x \ z) \ (y \ z) ) holds the carrier of X = BCK-part X
t3_bciideal:: for X being BCI-algebra for x, y, z, u, v being Element of X holds (x \ (y \ (z \ u))) \ (x \ (y \ (z \ v))) <= v \ u
t30_bciideal:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds (x \ y) \ (y \ x) = x \ y ) holds the carrier of X = BCK-part X
t31_bciideal:: for X being BCI-algebra st ( for X being BCI-algebra for x, y being Element of X holds (x \ y) \ ((x \ y) \ (y \ x)) = 0. X ) holds the carrier of X = BCK-part X
t32_bciideal:: for X being BCK-algebra holds the carrier of X is commutative Ideal of X
t33_bciideal:: for X being BCK-algebra for I being Ideal of X holds ( I is commutative Ideal of X iff for x, y being Element of X st x \ y in I holds x \ (y \ (y \ x)) in I )
t34_bciideal:: for X being BCK-algebra for I, A being Ideal of X st I c= A & I is commutative Ideal of X holds A is commutative Ideal of X
t35_bciideal:: for X being BCK-algebra holds ( ( for I being Ideal of X holds I is commutative Ideal of X ) iff {(0. X)} is commutative Ideal of X )
t36_bciideal:: for X being BCK-algebra holds ( {(0. X)} is commutative Ideal of X iff X is commutative BCK-algebra )
t37_bciideal:: for X being BCK-algebra holds ( X is commutative BCK-algebra iff for I being Ideal of X holds I is commutative Ideal of X )
t38_bciideal:: for X being BCK-algebra holds ( {(0. X)} is commutative Ideal of X iff for I being Ideal of X holds I is commutative Ideal of X )
t39_bciideal:: for X being BCK-algebra for I being Ideal of X for x, y being Element of X st x \ (x \ y) in I holds ( x \ ((x \ y) \ ((x \ y) \ x)) in I & (y \ (y \ x)) \ x in I & (y \ (y \ x)) \ (x \ y) in I )
t4_bciideal:: for X being BCI-algebra for x, y being Element of X holds ((0. X) \ (x \ y)) \ (y \ x) = 0. X
t40_bciideal:: for X being BCK-algebra holds ( {(0. X)} is commutative Ideal of X iff for x, y being Element of X holds x \ (x \ y) <= y \ (y \ x) )
t41_bciideal:: for X being BCK-algebra holds ( {(0. X)} is commutative Ideal of X iff for x, y being Element of X holds x \ y = x \ (y \ (y \ x)) )
t42_bciideal:: for X being BCK-algebra holds ( {(0. X)} is commutative Ideal of X iff for x, y being Element of X holds x \ (x \ y) = y \ (y \ (x \ (x \ y))) )
t43_bciideal:: for X being BCK-algebra holds ( {(0. X)} is commutative Ideal of X iff for x, y being Element of X st x <= y holds x = y \ (y \ x) )
t44_bciideal:: for X being BCK-algebra st {(0. X)} is commutative Ideal of X holds ( ( for x, y being Element of X holds ( x \ y = x iff y \ (y \ x) = 0. X ) ) & ( for x, y being Element of X st x \ y = x holds y \ x = y ) & ( for x, y, a being Element of X st y <= a holds (a \ x) \ (a \ y) = y \ x ) & ( for x, y being Element of X holds ( x \ (y \ (y \ x)) = x \ y & (x \ y) \ ((x \ y) \ x) = x \ y ) ) & ( for x, y, a being Element of X st x <= a holds (a \ y) \ ((a \ y) \ (a \ x)) = (a \ y) \ (x \ y) ) )
t45_bciideal:: for X being BCK-algebra holds ( ( for I being Ideal of X holds I is commutative Ideal of X ) iff for x, y being Element of X holds x \ (x \ y) <= y \ (y \ x) )
t46_bciideal:: for X being BCK-algebra holds ( ( for I being Ideal of X holds I is commutative Ideal of X ) iff for x, y being Element of X holds x \ y = x \ (y \ (y \ x)) )
t47_bciideal:: for X being BCK-algebra holds ( ( for I being Ideal of X holds I is commutative Ideal of X ) iff for x, y being Element of X holds x \ (x \ y) = y \ (y \ (x \ (x \ y))) )
t48_bciideal:: for X being BCK-algebra holds ( ( for I being Ideal of X holds I is commutative Ideal of X ) iff for x, y being Element of X st x <= y holds x = y \ (y \ x) )
t49_bciideal:: for X being BCK-algebra st ( for I being Ideal of X holds I is commutative Ideal of X ) holds ( ( for x, y being Element of X holds ( x \ y = x iff y \ (y \ x) = 0. X ) ) & ( for x, y being Element of X st x \ y = x holds y \ x = y ) & ( for x, y, a being Element of X st y <= a holds (a \ x) \ (a \ y) = y \ x ) & ( for x, y being Element of X holds ( x \ (y \ (y \ x)) = x \ y & (x \ y) \ ((x \ y) \ x) = x \ y ) ) & ( for x, y, a being Element of X st x <= a holds (a \ y) \ ((a \ y) \ (a \ x)) = (a \ y) \ (x \ y) ) )
t5_bciideal:: for X being BCI-algebra for A being Ideal of X for x being Element of X for a being Element of A st x <= a holds x in A
t50_bciideal:: for X being BCK-algebra for I being Ideal of X holds ( I is implicative-ideal of X iff for x, y being Element of X st x \ (y \ x) in I holds x in I )
t51_bciideal:: for X being BCK-algebra for I being Ideal of X holds ( I is positive-implicative-ideal of X iff for x, y being Element of X st (x \ y) \ y in I holds x \ y in I )
t52_bciideal:: for X being BCK-algebra for I being Ideal of X st ( for x, y, z being Element of X st (x \ y) \ z in I & y \ z in I holds x \ z in I ) holds for x, y, z being Element of X st (x \ y) \ z in I holds (x \ z) \ (y \ z) in I
t53_bciideal:: for X being BCK-algebra for I being Ideal of X st ( for x, y, z being Element of X st (x \ y) \ z in I holds (x \ z) \ (y \ z) in I ) holds I is positive-implicative-ideal of X
t54_bciideal:: for X being BCK-algebra for I being Ideal of X holds ( I is positive-implicative-ideal of X iff for x, y, z being Element of X st (x \ y) \ z in I & y \ z in I holds x \ z in I )
t55_bciideal:: for X being BCK-algebra for I being Ideal of X holds ( I is positive-implicative-ideal of X iff for x, y, z being Element of X st (x \ y) \ z in I holds (x \ z) \ (y \ z) in I )
t56_bciideal:: for X being BCK-algebra for I, A being Ideal of X st I c= A & I is positive-implicative-ideal of X holds A is positive-implicative-ideal of X
t57_bciideal:: for X being BCK-algebra for I being Ideal of X st I is implicative-ideal of X holds ( I is commutative Ideal of X & I is positive-implicative-ideal of X )
t6_bciideal:: for X being BCI-algebra for x, a, b being Element of AtomSet X st x is Element of BranchV b holds a \ x = a \ b
t7_bciideal:: for X being BCI-algebra for a being Element of X for x, b being Element of AtomSet X st x is Element of BranchV b holds a \ x = a \ b
t8_bciideal:: for X being BCI-algebra for A being Ideal of X for a being Element of A holds initial_section a c= A
t9_bciideal:: for X being BCI-algebra st AtomSet X is Ideal of X holds for x being Element of BCK-part X for a being Element of AtomSet X st x \ a in AtomSet X holds x = 0. X
d1_bhsp_1:: for X being non empty UNITSTR for x, y being Point of X holds x .|. y = the scalar of X . [x,y];
d2_bhsp_1:: for IT being non empty UNITSTR holds ( IT is RealUnitarySpace-like iff for x, y, z being Point of IT for a being Real holds ( ( x .|. x = 0 implies x = 0. IT ) & ( x = 0. IT implies x .|. x = 0 ) & 0 <= x .|. x & x .|. y = y .|. x & (x + y) .|. z = (x .|. z) + (y .|. z) & (a * x) .|. y = a * (x .|. y) ) );
d3_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds ( x,y are_orthogonal iff x .|. y = 0 );
d4_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds ||.x.|| = sqrt (x .|. x);
d5_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds dist (x,y) = ||.(x - y).||;
d6_bhsp_1:: for X being non empty addLoopStr for seq being sequence of X for x being Point of X for b4 being sequence of X holds ( b4 = seq + x iff for n being Element of NAT holds b4 . n = (seq . n) + x );
t1_bhsp_1:: for X being RealUnitarySpace holds (0. X) .|. (0. X) = 0 by Def2;
t10_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds (- x) .|. (- y) = x .|. y
t11_bhsp_1:: for X being RealUnitarySpace for x, y, z being Point of X holds (x - y) .|. z = (x .|. z) - (y .|. z)
t12_bhsp_1:: for X being RealUnitarySpace for x, y, z being Point of X holds x .|. (y - z) = (x .|. y) - (x .|. z)
t13_bhsp_1:: for X being RealUnitarySpace for x, y, u, v being Point of X holds (x - y) .|. (u - v) = (((x .|. u) - (x .|. v)) - (y .|. u)) + (y .|. v)
t14_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds (0. X) .|. x = 0
t15_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds x .|. (0. X) = 0 by Th14;
t16_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds (x + y) .|. (x + y) = ((x .|. x) + (2 * (x .|. y))) + (y .|. y)
t17_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds (x + y) .|. (x - y) = (x .|. x) - (y .|. y)
t18_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds (x - y) .|. (x - y) = ((x .|. x) - (2 * (x .|. y))) + (y .|. y)
t19_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds abs (x .|. y) <= (sqrt (x .|. x)) * (sqrt (y .|. y))
t2_bhsp_1:: for X being RealUnitarySpace for x, y, z being Point of X holds x .|. (y + z) = (x .|. y) + (x .|. z) by Def2;
t20_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X st x,y are_orthogonal holds x, - y are_orthogonal
t21_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X st x,y are_orthogonal holds - x,y are_orthogonal
t22_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X st x,y are_orthogonal holds - x, - y are_orthogonal
t23_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds x, 0. X are_orthogonal
t24_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X st x,y are_orthogonal holds (x + y) .|. (x + y) = (x .|. x) + (y .|. y)
t25_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X st x,y are_orthogonal holds (x - y) .|. (x - y) = (x .|. x) + (y .|. y)
t26_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds ( ||.x.|| = 0 iff x = 0. X )
t27_bhsp_1:: for a being Real for X being RealUnitarySpace for x being Point of X holds ||.(a * x).|| = (abs a) * ||.x.||
t28_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds 0 <= ||.x.||
t29_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds abs (x .|. y) <= ||.x.|| * ||.y.|| by Th19;
t3_bhsp_1:: for a being Real for X being RealUnitarySpace for x, y being Point of X holds x .|. (a * y) = a * (x .|. y) by Def2;
t30_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds ||.(x + y).|| <= ||.x.|| + ||.y.||
t31_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds ||.(- x).|| = ||.x.||
t32_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds ||.x.|| - ||.y.|| <= ||.(x - y).||
t33_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds abs (||.x.|| - ||.y.||) <= ||.(x - y).||
t34_bhsp_1:: for X being RealUnitarySpace for x being Point of X holds dist (x,x) = 0
t35_bhsp_1:: for X being RealUnitarySpace for x, z, y being Point of X holds dist (x,z) <= (dist (x,y)) + (dist (y,z))
t36_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds ( x <> y iff dist (x,y) <> 0 )
t37_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds dist (x,y) >= 0 by Th28;
t38_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds ( x <> y iff dist (x,y) > 0 )
t39_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds dist (x,y) = sqrt ((x - y) .|. (x - y)) ;
t4_bhsp_1:: for a being Real for X being RealUnitarySpace for x, y being Point of X holds (a * x) .|. y = x .|. (a * y)
t40_bhsp_1:: for X being RealUnitarySpace for x, y, u, v being Point of X holds dist ((x + y),(u + v)) <= (dist (x,u)) + (dist (y,v))
t41_bhsp_1:: for X being RealUnitarySpace for x, y, u, v being Point of X holds dist ((x - y),(u - v)) <= (dist (x,u)) + (dist (y,v))
t42_bhsp_1:: for X being RealUnitarySpace for x, z, y being Point of X holds dist ((x - z),(y - z)) = dist (x,y)
t43_bhsp_1:: for X being RealUnitarySpace for x, z, y being Point of X holds dist ((x - z),(y - z)) <= (dist (z,x)) + (dist (z,y))
t44_bhsp_1:: for n being Element of NAT for X being non empty addLoopStr for seq being sequence of X holds (- seq) . n = - (seq . n)
t45_bhsp_1:: for X being RealUnitarySpace for seq1, seq2, seq3 being sequence of X holds seq1 + (seq2 + seq3) = (seq1 + seq2) + seq3
t46_bhsp_1:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is constant & seq2 is constant holds seq1 + seq2 is constant
t47_bhsp_1:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is constant & seq2 is constant holds seq1 - seq2 is constant
t48_bhsp_1:: for a being Real for X being RealUnitarySpace for seq1 being sequence of X st seq1 is constant holds a * seq1 is constant
t49_bhsp_1:: for X being RealUnitarySpace for seq1, seq2 being sequence of X holds seq1 - seq2 = seq1 + (- seq2)
t5_bhsp_1:: for a, b being Real for X being RealUnitarySpace for x, y, z being Point of X holds ((a * x) + (b * y)) .|. z = (a * (x .|. z)) + (b * (y .|. z))
t50_bhsp_1:: for X being RealUnitarySpace for seq being sequence of X holds seq = seq + (0. X)
t51_bhsp_1:: for a being Real for X being RealUnitarySpace for seq1, seq2 being sequence of X holds a * (seq1 + seq2) = (a * seq1) + (a * seq2)
t52_bhsp_1:: for a, b being Real for X being RealUnitarySpace for seq being sequence of X holds (a + b) * seq = (a * seq) + (b * seq)
t53_bhsp_1:: for a, b being Real for X being RealUnitarySpace for seq being sequence of X holds (a * b) * seq = a * (b * seq)
t54_bhsp_1:: for X being RealUnitarySpace for seq being sequence of X holds 1 * seq = seq
t55_bhsp_1:: for X being RealUnitarySpace for seq being sequence of X holds (- 1) * seq = - seq
t56_bhsp_1:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X holds seq - x = seq + (- x)
t57_bhsp_1:: for X being RealUnitarySpace for seq1, seq2 being sequence of X holds seq1 - seq2 = - (seq2 - seq1)
t58_bhsp_1:: for X being RealUnitarySpace for seq being sequence of X holds seq = seq - (0. X)
t59_bhsp_1:: for X being RealUnitarySpace for seq being sequence of X holds seq = - (- seq)
t6_bhsp_1:: for a, b being Real for X being RealUnitarySpace for x, y, z being Point of X holds x .|. ((a * y) + (b * z)) = (a * (x .|. y)) + (b * (x .|. z)) by Th5;
t60_bhsp_1:: for X being RealUnitarySpace for seq1, seq2, seq3 being sequence of X holds seq1 - (seq2 + seq3) = (seq1 - seq2) - seq3
t61_bhsp_1:: for X being RealUnitarySpace for seq1, seq2, seq3 being sequence of X holds (seq1 + seq2) - seq3 = seq1 + (seq2 - seq3)
t62_bhsp_1:: for X being RealUnitarySpace for seq1, seq2, seq3 being sequence of X holds seq1 - (seq2 - seq3) = (seq1 - seq2) + seq3
t63_bhsp_1:: for a being Real for X being RealUnitarySpace for seq1, seq2 being sequence of X holds a * (seq1 - seq2) = (a * seq1) - (a * seq2)
t7_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds (- x) .|. y = x .|. (- y)
t8_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds (- x) .|. y = - (x .|. y)
t9_bhsp_1:: for X being RealUnitarySpace for x, y being Point of X holds x .|. (- y) = - (x .|. y) by Th8;
d1_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X holds ( seq is convergent iff ex g being Point of X st for r being Real st r > 0 holds ex m being Element of NAT st for n being Element of NAT st n >= m holds dist ((seq . n),g) < r );
d2_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X st seq is convergent holds for b3 being Point of X holds ( b3 = lim seq iff for r being Real st r > 0 holds ex m being Element of NAT st for n being Element of NAT st n >= m holds dist ((seq . n),b3) < r );
d3_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X for b3 being Real_Sequence holds ( b3 = ||.seq.|| iff for n being Element of NAT holds b3 . n = ||.(seq . n).|| );
d4_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X for x being Point of X for b4 being Real_Sequence holds ( b4 = dist (seq,x) iff for n being Element of NAT holds b4 . n = dist ((seq . n),x) );
d5_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real holds Ball (x,r) = { y where y is Point of X : ||.(x - y).|| < r } ;
d6_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real holds cl_Ball (x,r) = { y where y is Point of X : ||.(x - y).|| <= r } ;
d7_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real holds Sphere (x,r) = { y where y is Point of X : ||.(x - y).|| = r } ;
t1_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X st seq is constant holds seq is convergent
t10_bhsp_2:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is constant & x in rng seq holds lim seq = x
t11_bhsp_2:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is constant & ex n being Element of NAT st seq . n = x holds lim seq = x
t12_bhsp_2:: for X being RealUnitarySpace for seq, seq9 being sequence of X st seq is convergent & ex k being Element of NAT st for n being Element of NAT st n >= k holds seq9 . n = seq . n holds lim seq = lim seq9
t13_bhsp_2:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds lim (seq1 + seq2) = (lim seq1) + (lim seq2)
t14_bhsp_2:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds lim (seq1 - seq2) = (lim seq1) - (lim seq2)
t15_bhsp_2:: for X being RealUnitarySpace for a being Real for seq being sequence of X st seq is convergent holds lim (a * seq) = a * (lim seq)
t16_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X st seq is convergent holds lim (- seq) = - (lim seq)
t17_bhsp_2:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is convergent holds lim (seq + x) = (lim seq) + x
t18_bhsp_2:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is convergent holds lim (seq - x) = (lim seq) - x
t19_bhsp_2:: for X being RealUnitarySpace for g being Point of X for seq being sequence of X st seq is convergent holds ( lim seq = g iff for r being Real st r > 0 holds ex m being Element of NAT st for n being Element of NAT st n >= m holds ||.((seq . n) - g).|| < r )
t2_bhsp_2:: for X being RealUnitarySpace for seq, seq9 being sequence of X st seq is convergent & ex k being Element of NAT st for n being Element of NAT st k <= n holds seq9 . n = seq . n holds seq9 is convergent
t20_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X st seq is convergent holds ||.seq.|| is convergent
t21_bhsp_2:: for X being RealUnitarySpace for g being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.seq.|| is convergent & lim ||.seq.|| = ||.g.|| )
t22_bhsp_2:: for X being RealUnitarySpace for g being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.(seq - g).|| is convergent & lim ||.(seq - g).|| = 0 )
t23_bhsp_2:: for X being RealUnitarySpace for g being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds dist (seq,g) is convergent
t24_bhsp_2:: for X being RealUnitarySpace for g being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 )
t25_bhsp_2:: for X being RealUnitarySpace for g1, g2 being Point of X for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds ( ||.(seq1 + seq2).|| is convergent & lim ||.(seq1 + seq2).|| = ||.(g1 + g2).|| )
t26_bhsp_2:: for X being RealUnitarySpace for g1, g2 being Point of X for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds ( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )
t27_bhsp_2:: for X being RealUnitarySpace for g1, g2 being Point of X for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds ( ||.(seq1 - seq2).|| is convergent & lim ||.(seq1 - seq2).|| = ||.(g1 - g2).|| )
t28_bhsp_2:: for X being RealUnitarySpace for g1, g2 being Point of X for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds ( ||.((seq1 - seq2) - (g1 - g2)).|| is convergent & lim ||.((seq1 - seq2) - (g1 - g2)).|| = 0 )
t29_bhsp_2:: for X being RealUnitarySpace for g being Point of X for a being Real for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.(a * seq).|| is convergent & lim ||.(a * seq).|| = ||.(a * g).|| )
t3_bhsp_2:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds seq1 + seq2 is convergent
t30_bhsp_2:: for X being RealUnitarySpace for g being Point of X for a being Real for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )
t31_bhsp_2:: for X being RealUnitarySpace for g being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.(- seq).|| is convergent & lim ||.(- seq).|| = ||.(- g).|| )
t32_bhsp_2:: for X being RealUnitarySpace for g being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.((- seq) - (- g)).|| is convergent & lim ||.((- seq) - (- g)).|| = 0 )
t33_bhsp_2:: for X being RealUnitarySpace for g, x being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )
t34_bhsp_2:: for X being RealUnitarySpace for g, x being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.(seq - x).|| is convergent & lim ||.(seq - x).|| = ||.(g - x).|| )
t35_bhsp_2:: for X being RealUnitarySpace for g, x being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( ||.((seq - x) - (g - x)).|| is convergent & lim ||.((seq - x) - (g - x)).|| = 0 )
t36_bhsp_2:: for X being RealUnitarySpace for g1, g2 being Point of X for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds ( dist ((seq1 + seq2),(g1 + g2)) is convergent & lim (dist ((seq1 + seq2),(g1 + g2))) = 0 )
t37_bhsp_2:: for X being RealUnitarySpace for g1, g2 being Point of X for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds ( dist ((seq1 - seq2),(g1 - g2)) is convergent & lim (dist ((seq1 - seq2),(g1 - g2))) = 0 )
t38_bhsp_2:: for X being RealUnitarySpace for g being Point of X for a being Real for seq being sequence of X st seq is convergent & lim seq = g holds ( dist ((a * seq),(a * g)) is convergent & lim (dist ((a * seq),(a * g))) = 0 )
t39_bhsp_2:: for X being RealUnitarySpace for g, x being Point of X for seq being sequence of X st seq is convergent & lim seq = g holds ( dist ((seq + x),(g + x)) is convergent & lim (dist ((seq + x),(g + x))) = 0 )
t4_bhsp_2:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds seq1 - seq2 is convergent
t40_bhsp_2:: for X being RealUnitarySpace for z, x being Point of X for r being Real holds ( z in Ball (x,r) iff ||.(x - z).|| < r )
t41_bhsp_2:: for X being RealUnitarySpace for z, x being Point of X for r being Real holds ( z in Ball (x,r) iff dist (x,z) < r )
t42_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real st r > 0 holds x in Ball (x,r)
t43_bhsp_2:: for X being RealUnitarySpace for y, x, z being Point of X for r being Real st y in Ball (x,r) & z in Ball (x,r) holds dist (y,z) < 2 * r
t44_bhsp_2:: for X being RealUnitarySpace for y, x, z being Point of X for r being Real st y in Ball (x,r) holds y - z in Ball ((x - z),r)
t45_bhsp_2:: for X being RealUnitarySpace for y, x being Point of X for r being Real st y in Ball (x,r) holds y - x in Ball ((0. X),r)
t46_bhsp_2:: for X being RealUnitarySpace for y, x being Point of X for r, q being Real st y in Ball (x,r) & r <= q holds y in Ball (x,q)
t47_bhsp_2:: for X being RealUnitarySpace for z, x being Point of X for r being Real holds ( z in cl_Ball (x,r) iff ||.(x - z).|| <= r )
t48_bhsp_2:: for X being RealUnitarySpace for z, x being Point of X for r being Real holds ( z in cl_Ball (x,r) iff dist (x,z) <= r )
t49_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real st r >= 0 holds x in cl_Ball (x,r)
t5_bhsp_2:: for X being RealUnitarySpace for a being Real for seq being sequence of X st seq is convergent holds a * seq is convergent
t50_bhsp_2:: for X being RealUnitarySpace for y, x being Point of X for r being Real st y in Ball (x,r) holds y in cl_Ball (x,r)
t51_bhsp_2:: for X being RealUnitarySpace for z, x being Point of X for r being Real holds ( z in Sphere (x,r) iff ||.(x - z).|| = r )
t52_bhsp_2:: for X being RealUnitarySpace for z, x being Point of X for r being Real holds ( z in Sphere (x,r) iff dist (x,z) = r )
t53_bhsp_2:: for X being RealUnitarySpace for y, x being Point of X for r being Real st y in Sphere (x,r) holds y in cl_Ball (x,r)
t54_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real holds Ball (x,r) c= cl_Ball (x,r)
t55_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real holds Sphere (x,r) c= cl_Ball (x,r)
t56_bhsp_2:: for X being RealUnitarySpace for x being Point of X for r being Real holds (Ball (x,r)) \/ (Sphere (x,r)) = cl_Ball (x,r)
t6_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X st seq is convergent holds - seq is convergent
t7_bhsp_2:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is convergent holds seq + x is convergent
t8_bhsp_2:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is convergent holds seq - x is convergent
t9_bhsp_2:: for X being RealUnitarySpace for seq being sequence of X holds ( seq is convergent iff ex g being Point of X st for r being Real st r > 0 holds ex m being Element of NAT st for n being Element of NAT st n >= m holds ||.((seq . n) - g).|| < r )
d1_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X holds ( seq is Cauchy iff for r being Real st r > 0 holds ex k being Element of NAT st for n, m being Element of NAT st n >= k & m >= k holds dist ((seq . n),(seq . m)) < r );
d2_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X holds ( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds ex m being Element of NAT st for n being Element of NAT st n >= m holds dist ((seq1 . n),(seq2 . n)) < r );
d3_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X holds ( seq is bounded iff ex M being Real st ( M > 0 & ( for n being Element of NAT holds ||.(seq . n).|| <= M ) ) );
d4_bhsp_3:: for X being RealUnitarySpace holds ( X is complete iff for seq being sequence of X st seq is Cauchy holds seq is convergent );
t1_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X st seq is constant holds seq is Cauchy
t10_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X holds seq is_compared_to seq
t11_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is_compared_to seq2 holds seq2 is_compared_to seq1
t12_bhsp_3:: for X being RealUnitarySpace for seq1, seq2, seq3 being sequence of X st seq1 is_compared_to seq2 & seq2 is_compared_to seq3 holds seq1 is_compared_to seq3
t13_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X holds ( seq1 is_compared_to seq2 iff for r being Real st r > 0 holds ex m being Element of NAT st for n being Element of NAT st n >= m holds ||.((seq1 . n) - (seq2 . n)).|| < r )
t14_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st ex k being Element of NAT st for n being Element of NAT st n >= k holds seq1 . n = seq2 . n holds seq1 is_compared_to seq2
t15_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is Cauchy & seq1 is_compared_to seq2 holds seq2 is Cauchy
t16_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is convergent & seq1 is_compared_to seq2 holds seq2 is convergent
t17_bhsp_3:: for X being RealUnitarySpace for g being Point of X for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g & seq1 is_compared_to seq2 holds ( seq2 is convergent & lim seq2 = g )
t18_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds seq1 + seq2 is bounded
t19_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X st seq is bounded holds - seq is bounded
t2_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X holds ( seq is Cauchy iff for r being Real st r > 0 holds ex k being Element of NAT st for n, m being Element of NAT st n >= k & m >= k holds ||.((seq . n) - (seq . m)).|| < r )
t20_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is bounded & seq2 is bounded holds seq1 - seq2 is bounded
t21_bhsp_3:: for X being RealUnitarySpace for a being Real for seq being sequence of X st seq is bounded holds a * seq is bounded
t22_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X st seq is constant holds seq is bounded
t23_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for m being Element of NAT ex M being Real st ( M > 0 & ( for n being Element of NAT st n <= m holds ||.(seq . n).|| < M ) )
t24_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X st seq is convergent holds seq is bounded
t25_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is bounded & seq1 is_compared_to seq2 holds seq2 is bounded
t26_bhsp_3:: for X being RealUnitarySpace for seq, seq1 being sequence of X st seq is bounded & seq1 is subsequence of seq holds seq1 is bounded
t27_bhsp_3:: for X being RealUnitarySpace for seq, seq1 being sequence of X st seq is convergent & seq1 is subsequence of seq holds seq1 is convergent
t28_bhsp_3:: for X being RealUnitarySpace for seq1, seq being sequence of X st seq1 is subsequence of seq & seq is convergent holds lim seq1 = lim seq
t29_bhsp_3:: for X being RealUnitarySpace for seq, seq1 being sequence of X st seq is Cauchy & seq1 is subsequence of seq holds seq1 is Cauchy
t3_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds seq1 + seq2 is Cauchy
t30_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for k, m being Element of NAT holds (seq ^\ k) ^\ m = (seq ^\ m) ^\ k
t31_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for k, m being Element of NAT holds (seq ^\ k) ^\ m = seq ^\ (k + m)
t32_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X for k being Element of NAT holds (seq1 + seq2) ^\ k = (seq1 ^\ k) + (seq2 ^\ k)
t33_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for k being Element of NAT holds (- seq) ^\ k = - (seq ^\ k)
t34_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X for k being Element of NAT holds (seq1 - seq2) ^\ k = (seq1 ^\ k) - (seq2 ^\ k)
t35_bhsp_3:: for X being RealUnitarySpace for a being Real for seq being sequence of X for k being Element of NAT holds (a * seq) ^\ k = a * (seq ^\ k)
t36_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for k being Element of NAT st seq is convergent holds ( seq ^\ k is convergent & lim (seq ^\ k) = lim seq ) by Th27, Th28;
t37_bhsp_3:: for X being RealUnitarySpace for seq, seq1 being sequence of X st seq is convergent & ex k being Element of NAT st seq = seq1 ^\ k holds seq1 is convergent
t38_bhsp_3:: for X being RealUnitarySpace for seq, seq1 being sequence of X st seq is Cauchy & ex k being Element of NAT st seq = seq1 ^\ k holds seq1 is Cauchy
t39_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for k being Element of NAT st seq is Cauchy holds seq ^\ k is Cauchy by Th29;
t4_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is Cauchy & seq2 is Cauchy holds seq1 - seq2 is Cauchy
t40_bhsp_3:: for X being RealUnitarySpace for seq1, seq2 being sequence of X for k being Element of NAT st seq1 is_compared_to seq2 holds seq1 ^\ k is_compared_to seq2 ^\ k
t41_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for k being Element of NAT st seq is bounded holds seq ^\ k is bounded by Th26;
t42_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X for k being Element of NAT st seq is constant holds seq ^\ k is constant ;
t43_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X st X is complete & seq is Cauchy holds seq is bounded
t5_bhsp_3:: for X being RealUnitarySpace for a being Real for seq being sequence of X st seq is Cauchy holds a * seq is Cauchy
t6_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X st seq is Cauchy holds - seq is Cauchy
t7_bhsp_3:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is Cauchy holds seq + x is Cauchy
t8_bhsp_3:: for X being RealUnitarySpace for x being Point of X for seq being sequence of X st seq is Cauchy holds seq - x is Cauchy
t9_bhsp_3:: for X being RealUnitarySpace for seq being sequence of X st seq is convergent holds seq is Cauchy
d1_bhsp_4:: for X being non empty addLoopStr for seq, b3 being sequence of X holds ( b3 = Partial_Sums seq iff ( b3 . 0 = seq . 0 & ( for n being Element of NAT holds b3 . (n + 1) = (b3 . n) + (seq . (n + 1)) ) ) );
d2_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds ( seq is summable iff Partial_Sums seq is convergent );
d3_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds Sum seq = lim (Partial_Sums seq);
d4_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n being Element of NAT holds Sum (seq,n) = (Partial_Sums seq) . n;
d5_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n, m being Element of NAT holds Sum (seq,n,m) = (Sum (seq,n)) - (Sum (seq,m));
d6_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds ( seq is absolutely_summable iff ||.seq.|| is summable );
d7_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for Rseq being Real_Sequence for b4 being sequence of X holds ( b4 = Rseq * seq iff for n being Element of NAT holds b4 . n = (Rseq . n) * (seq . n) );
d8_bhsp_4:: for Rseq being Real_Sequence holds ( Rseq is Cauchy iff for r being Real st r > 0 holds ex k being Element of NAT st for n, m being Element of NAT st n >= k & m >= k holds abs ((Rseq . n) - (Rseq . m)) < r );
t1_bhsp_4:: for X being non empty Abelian add-associative addLoopStr for seq1, seq2 being sequence of X holds (Partial_Sums seq1) + (Partial_Sums seq2) = Partial_Sums (seq1 + seq2)
t10_bhsp_4:: for X being RealHilbertSpace for seq being sequence of X holds ( seq is summable iff for r being Real st r > 0 holds ex k being Element of NAT st for n, m being Element of NAT st n >= k & m >= k holds ||.(((Partial_Sums seq) . n) - ((Partial_Sums seq) . m)).|| < r )
t11_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X st seq is summable holds Partial_Sums seq is bounded
t12_bhsp_4:: for X being RealUnitarySpace for seq, seq1 being sequence of X st ( for n being Element of NAT holds seq1 . n = seq . 0 ) holds Partial_Sums (seq ^\ 1) = ((Partial_Sums seq) ^\ 1) - seq1
t13_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X st seq is summable holds for k being Element of NAT holds seq ^\ k is summable
t14_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X st ex k being Element of NAT st seq ^\ k is summable holds seq is summable
t15_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds Sum (seq,0) = seq . 0 by Def1;
t16_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds Sum (seq,1) = (Sum (seq,0)) + (seq . 1)
t17_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds Sum (seq,1) = (seq . 0) + (seq . 1)
t18_bhsp_4:: for n being Element of NAT for X being RealUnitarySpace for seq being sequence of X holds Sum (seq,(n + 1)) = (Sum (seq,n)) + (seq . (n + 1)) by Def1;
t19_bhsp_4:: for n being Element of NAT for X being RealUnitarySpace for seq being sequence of X holds seq . (n + 1) = (Sum (seq,(n + 1))) - (Sum (seq,n))
t2_bhsp_4:: for X being non empty right_complementable Abelian add-associative right_zeroed addLoopStr for seq1, seq2 being sequence of X holds (Partial_Sums seq1) - (Partial_Sums seq2) = Partial_Sums (seq1 - seq2)
t20_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds seq . 1 = (Sum (seq,1)) - (Sum (seq,0))
t21_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds Sum (seq,1,0) = seq . 1
t22_bhsp_4:: for n being Element of NAT for X being RealUnitarySpace for seq being sequence of X holds Sum (seq,(n + 1),n) = seq . (n + 1) by Th19;
t23_bhsp_4:: for X being RealHilbertSpace for seq being sequence of X holds ( seq is summable iff for r being Real st r > 0 holds ex k being Element of NAT st for n, m being Element of NAT st n >= k & m >= k holds ||.((Sum (seq,n)) - (Sum (seq,m))).|| < r )
t24_bhsp_4:: for X being RealHilbertSpace for seq being sequence of X holds ( seq is summable iff for r being Real st r > 0 holds ex k being Element of NAT st for n, m being Element of NAT st n >= k & m >= k holds ||.(Sum (seq,n,m)).|| < r )
t25_bhsp_4:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is absolutely_summable & seq2 is absolutely_summable holds seq1 + seq2 is absolutely_summable
t26_bhsp_4:: for a being Real for X being RealUnitarySpace for seq being sequence of X st seq is absolutely_summable holds a * seq is absolutely_summable
t27_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st ( for n being Element of NAT holds ||.seq.|| . n <= Rseq . n ) & Rseq is summable holds seq is absolutely_summable
t28_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st ( for n being Element of NAT holds ( seq . n <> 0. X & Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) ) & Rseq is convergent & lim Rseq < 1 holds seq is absolutely_summable
t29_bhsp_4:: for r being Real for X being RealUnitarySpace for seq being sequence of X st r > 0 & ex m being Element of NAT st for n being Element of NAT st n >= m holds ||.(seq . n).|| >= r & seq is convergent holds lim seq <> 0. X
t3_bhsp_4:: for a being Real for X being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct for seq being sequence of X holds Partial_Sums (a * seq) = a * (Partial_Sums seq)
t30_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X st ( for n being Element of NAT holds seq . n <> 0. X ) & ex m being Element of NAT st for n being Element of NAT st n >= m holds ||.(seq . (n + 1)).|| / ||.(seq . n).|| >= 1 holds not seq is summable
t31_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st ( for n being Element of NAT holds seq . n <> 0. X ) & ( for n being Element of NAT holds Rseq . n = ||.(seq . (n + 1)).|| / ||.(seq . n).|| ) & Rseq is convergent & lim Rseq > 1 holds not seq is summable
t32_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st ( for n being Element of NAT holds Rseq . n = n -root ||.(seq . n).|| ) & Rseq is convergent & lim Rseq < 1 holds seq is absolutely_summable
t33_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st ( for n being Element of NAT holds Rseq . n = n -root (||.seq.|| . n) ) & ex m being Element of NAT st for n being Element of NAT st n >= m holds Rseq . n >= 1 holds not seq is summable
t34_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st ( for n being Element of NAT holds Rseq . n = n -root (||.seq.|| . n) ) & Rseq is convergent & lim Rseq > 1 holds not seq is summable
t35_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds Partial_Sums ||.seq.|| is non-decreasing
t36_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n being Element of NAT holds (Partial_Sums ||.seq.||) . n >= 0
t37_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n being Element of NAT holds ||.((Partial_Sums seq) . n).|| <= (Partial_Sums ||.seq.||) . n
t38_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n being Element of NAT holds ||.(Sum (seq,n)).|| <= Sum (||.seq.||,n) by Th37;
t39_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n, m being Element of NAT holds ||.(((Partial_Sums seq) . m) - ((Partial_Sums seq) . n)).|| <= abs (((Partial_Sums ||.seq.||) . m) - ((Partial_Sums ||.seq.||) . n))
t4_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X holds Partial_Sums (- seq) = - (Partial_Sums seq)
t40_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n, m being Element of NAT holds ||.((Sum (seq,m)) - (Sum (seq,n))).|| <= abs ((Sum (||.seq.||,m)) - (Sum (||.seq.||,n))) by Th39;
t41_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X for n, m being Element of NAT holds ||.(Sum (seq,m,n)).|| <= abs (Sum (||.seq.||,m,n)) by Th39;
t42_bhsp_4:: for X being RealHilbertSpace for seq being sequence of X st seq is absolutely_summable holds seq is summable
t43_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq1, seq2 being sequence of X holds Rseq * (seq1 + seq2) = (Rseq * seq1) + (Rseq * seq2)
t44_bhsp_4:: for Rseq1, Rseq2 being Real_Sequence for X being RealUnitarySpace for seq being sequence of X holds (Rseq1 + Rseq2) * seq = (Rseq1 * seq) + (Rseq2 * seq)
t45_bhsp_4:: for Rseq1, Rseq2 being Real_Sequence for X being RealUnitarySpace for seq being sequence of X holds (Rseq1 (#) Rseq2) * seq = Rseq1 * (Rseq2 * seq)
t46_bhsp_4:: for a being Real for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X holds (a (#) Rseq) * seq = a * (Rseq * seq)
t47_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X holds Rseq * (- seq) = (- Rseq) * seq
t48_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st Rseq is convergent & seq is convergent holds Rseq * seq is convergent
t49_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st Rseq is bounded & seq is bounded holds Rseq * seq is bounded
t5_bhsp_4:: for a, b being Real for X being RealUnitarySpace for seq1, seq2 being sequence of X holds (a * (Partial_Sums seq1)) + (b * (Partial_Sums seq2)) = Partial_Sums ((a * seq1) + (b * seq2))
t50_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X st Rseq is convergent & seq is convergent holds ( Rseq * seq is convergent & lim (Rseq * seq) = (lim Rseq) * (lim seq) )
t51_bhsp_4:: for Rseq being Real_Sequence for X being RealHilbertSpace for seq being sequence of X st seq is Cauchy & Rseq is Cauchy holds Rseq * seq is Cauchy
t52_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X for n being Element of NAT holds (Partial_Sums ((Rseq - (Rseq ^\ 1)) * (Partial_Sums seq))) . n = ((Partial_Sums (Rseq * seq)) . (n + 1)) - ((Rseq * (Partial_Sums seq)) . (n + 1))
t53_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X for n being Element of NAT holds (Partial_Sums (Rseq * seq)) . (n + 1) = ((Rseq * (Partial_Sums seq)) . (n + 1)) - ((Partial_Sums (((Rseq ^\ 1) - Rseq) * (Partial_Sums seq))) . n)
t54_bhsp_4:: for Rseq being Real_Sequence for X being RealUnitarySpace for seq being sequence of X for n being Element of NAT holds Sum ((Rseq * seq),(n + 1)) = ((Rseq * (Partial_Sums seq)) . (n + 1)) - (Sum ((((Rseq ^\ 1) - Rseq) * (Partial_Sums seq)),n)) by Th53;
t6_bhsp_4:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is summable & seq2 is summable holds ( seq1 + seq2 is summable & Sum (seq1 + seq2) = (Sum seq1) + (Sum seq2) )
t7_bhsp_4:: for X being RealUnitarySpace for seq1, seq2 being sequence of X st seq1 is summable & seq2 is summable holds ( seq1 - seq2 is summable & Sum (seq1 - seq2) = (Sum seq1) - (Sum seq2) )
t8_bhsp_4:: for a being Real for X being RealUnitarySpace for seq being sequence of X st seq is summable holds ( a * seq is summable & Sum (a * seq) = a * (Sum seq) )
t9_bhsp_4:: for X being RealUnitarySpace for seq being sequence of X st seq is summable holds ( seq is convergent & lim seq = 0. X )
d1_bhsp_5:: for DX being non empty set for f being BinOp of DX st f is commutative & f is associative & f is having_a_unity holds for Y being finite Subset of DX for b4 being Element of DX holds ( b4 = f ++ Y iff ex p being FinSequence of DX st ( p is one-to-one & rng p = Y & b4 = f "**" p ) );
d2_bhsp_5:: for X being RealUnitarySpace for Y being finite Subset of X holds ( ( Y <> {} implies setop_SUM (Y,X) = the addF of X ++ Y ) & ( not Y <> {} implies setop_SUM (Y,X) = 0. X ) );
d3_bhsp_5:: for X being RealUnitarySpace for x being Point of X for p being FinSequence for i being Nat holds PO (i,p,x) = the scalar of X . [x,(p . i)];
d4_bhsp_5:: for DK, DX being non empty set for F being Function of DX,DK for p being FinSequence of DX holds Func_Seq (F,p) = F * p;
d5_bhsp_5:: for DK, DX being non empty set for f being BinOp of DK st f is commutative & f is associative & f is having_a_unity holds for Y being finite Subset of DX for F being Function of DX,DK st Y c= dom F holds for b6 being Element of DK holds ( b6 = setopfunc (Y,DX,DK,F,f) iff ex p being FinSequence of DX st ( p is one-to-one & rng p = Y & b6 = f "**" (Func_Seq (F,p)) ) );
d6_bhsp_5:: for X being RealUnitarySpace for x being Point of X for Y being finite Subset of X for b4 being Real holds ( b4 = setop_xPre_PROD (x,Y,X) iff ex p being FinSequence of the carrier of X st ( p is one-to-one & rng p = Y & ex q being FinSequence of REAL st ( dom q = dom p & ( for i being Element of NAT st i in dom q holds q . i = PO (i,p,x) ) & b4 = addreal "**" q ) ) );
d7_bhsp_5:: for X being RealUnitarySpace for x being Point of X for Y being finite Subset of X holds ( ( Y <> {} implies setop_xPROD (x,Y,X) = setop_xPre_PROD (x,Y,X) ) & ( not Y <> {} implies setop_xPROD (x,Y,X) = 0 ) );
d8_bhsp_5:: for X being RealUnitarySpace for b2 being Subset of X holds ( b2 is OrthogonalFamily of X iff for x, y being Point of X st x in b2 & y in b2 & x <> y holds x .|. y = 0 );
d9_bhsp_5:: for X being RealUnitarySpace for b2 being Subset of X holds ( b2 is OrthonormalFamily of X iff ( b2 is OrthogonalFamily of X & ( for x being Point of X st x in b2 holds x .|. x = 1 ) ) );
t1_bhsp_5:: for D being set for p1, p2 being FinSequence of D st p1 is one-to-one & p2 is one-to-one & rng p1 = rng p2 holds ( dom p1 = dom p2 & ex P being Permutation of (dom p1) st ( p2 = p1 * P & dom P = dom p1 & rng P = dom p1 ) )
t10_bhsp_5:: for X being RealUnitarySpace for x being Point of X for S being non empty finite Subset of X for F being Function of the carrier of X, the carrier of X st S c= dom F holds for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds H . y = the scalar of X . [x,(F . y)] ) holds for p being FinSequence of the carrier of X st p is one-to-one & rng p = S holds the scalar of X . [x,( the addF of X "**" (Func_Seq (F,p)))] = addreal "**" (Func_Seq (H,p))
t11_bhsp_5:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for x being Point of X for S being finite OrthonormalFamily of X st not S is empty holds for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds H . y = (x .|. y) ^2 ) holds for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y being Point of X st y in S holds F . y = (x .|. y) * y ) holds x .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
t12_bhsp_5:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for x being Point of X for S being finite OrthonormalFamily of X st not S is empty holds for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y being Point of X st y in S holds F . y = (x .|. y) * y ) holds for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds H . y = (x .|. y) ^2 ) holds (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
t13_bhsp_5:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for x being Point of X for S being finite OrthonormalFamily of X st not S is empty holds for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds H . y = (x .|. y) ^2 ) holds setopfunc (S, the carrier of X,REAL,H,addreal) <= ||.x.|| ^2
t14_bhsp_5:: for DK, DX being non empty set for f being BinOp of DK st f is commutative & f is associative & f is having_a_unity holds for Y1, Y2 being finite Subset of DX st Y1 misses Y2 holds for F being Function of DX,DK st Y1 c= dom F & Y2 c= dom F holds for Z being finite Subset of DX st Z = Y1 \/ Y2 holds setopfunc (Z,DX,DK,F,f) = f . ((setopfunc (Y1,DX,DK,F,f)),(setopfunc (Y2,DX,DK,F,f)))
t2_bhsp_5:: for X being RealUnitarySpace holds {} is OrthogonalFamily of X
t3_bhsp_5:: for X being RealUnitarySpace holds {} is OrthonormalFamily of X
t4_bhsp_5:: for X being RealUnitarySpace for x being Point of X holds ( x = 0. X iff for y being Point of X holds x .|. y = 0 )
t5_bhsp_5:: for X being RealUnitarySpace for x, y being Point of X holds (||.(x + y).|| ^2) + (||.(x - y).|| ^2) = (2 * (||.x.|| ^2)) + (2 * (||.y.|| ^2))
t6_bhsp_5:: for X being RealUnitarySpace for x, y being Point of X st x,y are_orthogonal holds ||.(x + y).|| ^2 = (||.x.|| ^2) + (||.y.|| ^2)
t7_bhsp_5:: for X being RealUnitarySpace for p being FinSequence of the carrier of X st len p >= 1 & ( for i, j being Element of NAT st i in dom p & j in dom p & i <> j holds the scalar of X . [(p . i),(p . j)] = 0 ) holds for q being FinSequence of REAL st dom p = dom q & ( for i being Element of NAT st i in dom q holds q . i = the scalar of X . [(p . i),(p . i)] ) holds ( the addF of X "**" p) .|. ( the addF of X "**" p) = addreal "**" q
t8_bhsp_5:: for X being RealUnitarySpace for x being Point of X for p being FinSequence of the carrier of X st len p >= 1 holds for q being FinSequence of REAL st dom p = dom q & ( for i being Element of NAT st i in dom q holds q . i = the scalar of X . [x,(p . i)] ) holds x .|. ( the addF of X "**" p) = addreal "**" q
t9_bhsp_5:: for X being RealUnitarySpace for S being non empty finite Subset of X for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y1, y2 being Point of X st y1 in S & y2 in S & y1 <> y2 holds the scalar of X . [(F . y1),(F . y2)] = 0 ) holds for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds H . y = the scalar of X . [(F . y),(F . y)] ) holds for p being FinSequence of the carrier of X st p is one-to-one & rng p = S holds the scalar of X . [( the addF of X "**" (Func_Seq (F,p))),( the addF of X "**" (Func_Seq (F,p)))] = addreal "**" (Func_Seq (H,p))
d1_bhsp_6:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for Y being finite Subset of X for b3 being Element of X holds ( b3 = setsum Y iff ex p being FinSequence of the carrier of X st ( p is one-to-one & rng p = Y & b3 = the addF of X "**" p ) );
d2_bhsp_6:: for X being RealUnitarySpace for Y being Subset of X holds ( Y is summable_set iff ex x being Point of X st for e being Real st e > 0 holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds ||.(x - (setsum Y1)).|| < e ) ) );
d3_bhsp_6:: for X being RealUnitarySpace for Y being Subset of X st Y is summable_set holds for b3 being Point of X holds ( b3 = sum Y iff for e being Real st e > 0 holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds ||.(b3 - (setsum Y1)).|| < e ) ) );
d4_bhsp_6:: for X being RealUnitarySpace for L being linear-Functional of X holds ( L is Lipschitzian iff ex K being Real st ( K > 0 & ( for x being Point of X holds abs (L . x) <= K * ||.x.|| ) ) );
d5_bhsp_6:: for X being RealUnitarySpace for Y being Subset of X holds ( Y is weakly_summable_set iff ex x being Point of X st for L being linear-Functional of X st L is Lipschitzian holds for e being Real st e > 0 holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs (L . (x - (setsum Y1))) < e ) ) );
d6_bhsp_6:: for X being RealUnitarySpace for Y being Subset of X for L being Functional of X holds ( Y is_summable_set_by L iff ex r being Real st for e being Real st e > 0 holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) ) );
d7_bhsp_6:: for X being RealUnitarySpace for Y being Subset of X for L being Functional of X st Y is_summable_set_by L holds for b4 being Real holds ( b4 = sum_byfunc (Y,L) iff for e being Real st e > 0 holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs (b4 - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) ) );
t1_bhsp_6:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for Y being finite Subset of X for I being Function of the carrier of X, the carrier of X st Y c= dom I & ( for x being set st x in dom I holds I . x = x ) holds setsum Y = setopfunc (Y, the carrier of X, the carrier of X,I, the addF of X)
t10_bhsp_6:: for X being RealHilbertSpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for Y being Subset of X holds ( Y is summable_set iff for e being Real st e > 0 holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds ||.(setsum Y1).|| < e ) ) )
t2_bhsp_6:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for Y1, Y2 being finite Subset of X st Y1 misses Y2 holds for Z being finite Subset of X st Z = Y1 \/ Y2 holds setsum Z = (setsum Y1) + (setsum Y2)
t3_bhsp_6:: for X being RealUnitarySpace for Y being Subset of X st Y is summable_set holds Y is weakly_summable_set
t4_bhsp_6:: for X being RealUnitarySpace for L being linear-Functional of X for p being FinSequence of the carrier of X st len p >= 1 holds for q being FinSequence of REAL st dom p = dom q & ( for i being Element of NAT st i in dom q holds q . i = L . (p . i) ) holds L . ( the addF of X "**" p) = addreal "**" q
t5_bhsp_6:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for S being finite Subset of X st not S is empty holds for L being linear-Functional of X holds L . (setsum S) = setopfunc (S, the carrier of X,REAL,L,addreal)
t6_bhsp_6:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for Y being Subset of X st Y is weakly_summable_set holds ex x being Point of X st for L being linear-Functional of X st L is Lipschitzian holds for e being Real st e > 0 holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds abs ((L . x) - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) )
t7_bhsp_6:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for Y being Subset of X st Y is weakly_summable_set holds for L being linear-Functional of X st L is Lipschitzian holds Y is_summable_set_by L
t8_bhsp_6:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for Y being Subset of X st Y is summable_set holds for L being linear-Functional of X st L is Lipschitzian holds Y is_summable_set_by L by Th3, Th7;
t9_bhsp_6:: for X being RealUnitarySpace for Y being finite Subset of X st not Y is empty holds Y is summable_set
t1_bhsp_7:: for X being RealUnitarySpace for Y being Subset of X for L being Functional of X holds ( Y is_summable_set_by L iff for e being Real st 0 < e holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) )
t2_bhsp_7:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for S being finite OrthogonalFamily of X st not S is empty holds for I being Function of the carrier of X, the carrier of X st S c= dom I & ( for y being Point of X st y in S holds I . y = y ) holds for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds H . y = y .|. y ) holds (setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,I, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
t3_bhsp_7:: for X being RealUnitarySpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for S being finite OrthogonalFamily of X st not S is empty holds for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds H . x = x .|. x ) holds (setsum S) .|. (setsum S) = setopfunc (S, the carrier of X,REAL,H,addreal)
t4_bhsp_7:: for X being RealUnitarySpace for Y being OrthogonalFamily of X for Z being Subset of X st Z is Subset of Y holds Z is OrthogonalFamily of X
t5_bhsp_7:: for X being RealUnitarySpace for Y being OrthonormalFamily of X for Z being Subset of X st Z is Subset of Y holds Z is OrthonormalFamily of X
t6_bhsp_7:: for X being RealHilbertSpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for S being OrthonormalFamily of X for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds H . x = x .|. x ) holds ( S is summable_set iff S is_summable_set_by H )
t7_bhsp_7:: for X being RealUnitarySpace for S being Subset of X st S is summable_set holds for e being Real st 0 < e holds ex Y0 being finite Subset of X st ( not Y0 is empty & Y0 c= S & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= S holds abs (((sum S) .|. (sum S)) - ((setsum Y1) .|. (setsum Y1))) < e ) )
t8_bhsp_7:: for X being RealHilbertSpace st the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity holds for S being OrthonormalFamily of X for H being Functional of X st S c= dom H & ( for x being Point of X st x in S holds H . x = x .|. x ) & S is summable_set holds (sum S) .|. (sum S) = sum_byfunc (S,H)
d1_bilinear:: for K being non empty ZeroStr for V, W being VectSpStr over K holds NulForm (V,W) = [: the carrier of V, the carrier of W:] --> (0. K);
d10_bilinear:: for K being non empty multMagma for V, W being non empty VectSpStr over K for f being Functional of V for g being Functional of W for b6 being Form of V,W holds ( b6 = FormFunctional (f,g) iff for v being Vector of V for w being Vector of W holds b6 . (v,w) = (f . v) * (g . w) );
d11_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W holds ( f is additiveFAF iff for v being Vector of V holds FunctionalFAF (f,v) is additive );
d12_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W holds ( f is additiveSAF iff for w being Vector of W holds FunctionalSAF (f,w) is additive );
d13_bilinear:: for K being non empty multMagma for V, W being non empty VectSpStr over K for f being Form of V,W holds ( f is homogeneousFAF iff for v being Vector of V holds FunctionalFAF (f,v) is homogeneous );
d14_bilinear:: for K being non empty multMagma for V, W being non empty VectSpStr over K for f being Form of V,W holds ( f is homogeneousSAF iff for w being Vector of W holds FunctionalSAF (f,w) is homogeneous );
d15_bilinear:: for K being ZeroStr for V, W being non empty VectSpStr over K for f being Form of V,W holds leftker f = { v where v is Vector of V : for w being Vector of W holds f . (v,w) = 0. K } ;
d16_bilinear:: for K being ZeroStr for V, W being non empty VectSpStr over K for f being Form of V,W holds rightker f = { w where w is Vector of W : for v being Vector of V holds f . (v,w) = 0. K } ;
d17_bilinear:: for K being ZeroStr for V being non empty VectSpStr over K for f being Form of V,V holds diagker f = { v where v is Vector of V : f . (v,v) = 0. K } ;
d18_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V being VectSp of K for W being non empty VectSpStr over K for f being additiveSAF homogeneousSAF Form of V,W for b5 being non empty strict Subspace of V holds ( b5 = LKer f iff the carrier of b5 = leftker f );
d19_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V being non empty VectSpStr over K for W being VectSp of K for f being additiveFAF homogeneousFAF Form of V,W for b5 being non empty strict Subspace of W holds ( b5 = RKer f iff the carrier of b5 = rightker f );
d2_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, g, b6 being Form of V,W holds ( b6 = f + g iff for v being Vector of V for w being Vector of W holds b6 . (v,w) = (f . (v,w)) + (g . (v,w)) );
d20_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V being VectSp of K for W being non empty VectSpStr over K for f being additiveSAF homogeneousSAF Form of V,W for b5 being additiveSAF homogeneousSAF Form of (VectQuot (V,(LKer f))),W holds ( b5 = LQForm f iff for A being Vector of (VectQuot (V,(LKer f))) for w being Vector of W for v being Vector of V st A = v + (LKer f) holds b5 . (A,w) = f . (v,w) );
d21_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V being non empty VectSpStr over K for W being VectSp of K for f being additiveFAF homogeneousFAF Form of V,W for b5 being additiveFAF homogeneousFAF Form of V,(VectQuot (W,(RKer f))) holds ( b5 = RQForm f iff for A being Vector of (VectQuot (W,(RKer f))) for v being Vector of V for w being Vector of W st A = w + (RKer f) holds b5 . (v,A) = f . (v,w) );
d22_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for f being bilinear-Form of V,W for b5 being bilinear-Form of (VectQuot (V,(LKer f))),(VectQuot (W,(RKer f))) holds ( b5 = QForm f iff for A being Vector of (VectQuot (V,(LKer f))) for B being Vector of (VectQuot (W,(RKer f))) for v being Vector of V for w being Vector of W st A = v + (LKer f) & B = w + (RKer f) holds b5 . (A,B) = f . (v,w) );
d23_bilinear:: for K being ZeroStr for V, W being non empty VectSpStr over K for f being Form of V,W holds ( f is degenerated-on-left iff leftker f <> {(0. V)} );
d24_bilinear:: for K being ZeroStr for V, W being non empty VectSpStr over K for f being Form of V,W holds ( f is degenerated-on-right iff rightker f <> {(0. W)} );
d25_bilinear:: for K being 1-sorted for V being VectSpStr over K for f being Form of V,V holds ( f is symmetric iff for v, w being Vector of V holds f . (v,w) = f . (w,v) );
d26_bilinear:: for K being ZeroStr for V being VectSpStr over K for f being Form of V,V holds ( f is alternating iff for v being Vector of V holds f . (v,v) = 0. K );
d27_bilinear:: for K being Fanoian Field for V being non empty VectSpStr over K for f being additiveFAF additiveSAF Form of V,V holds ( f is alternating iff for v, w being Vector of V holds f . (v,w) = - (f . (w,v)) );
d3_bilinear:: for K being non empty multMagma for V, W being non empty VectSpStr over K for f being Form of V,W for a being Element of K for b6 being Form of V,W holds ( b6 = a * f iff for v being Vector of V for w being Vector of W holds b6 . (v,w) = a * (f . (v,w)) );
d4_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, b5 being Form of V,W holds ( b5 = - f iff for v being Vector of V for w being Vector of W holds b5 . (v,w) = - (f . (v,w)) );
d5_bilinear:: for K being non empty right_complementable add-associative right_zeroed left-distributive left_unital doubleLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W holds - f = (- (1. K)) * f;
d6_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, g being Form of V,W holds f - g = f + (- g);
d7_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, g, b6 being Form of V,W holds ( b6 = f - g iff for v being Vector of V for w being Vector of W holds b6 . (v,w) = (f . (v,w)) - (g . (v,w)) );
d8_bilinear:: for K being non empty 1-sorted for V, W being non empty VectSpStr over K for f being Form of V,W for v being Vector of V holds FunctionalFAF (f,v) = (curry f) . v;
d9_bilinear:: for K being non empty 1-sorted for V, W being non empty VectSpStr over K for f being Form of V,W for w being Vector of W holds FunctionalSAF (f,w) = (curry' f) . w;
t1_bilinear:: for K being non empty right_zeroed addLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W holds f + (NulForm (V,W)) = f
t10_bilinear:: for K being non empty ZeroStr for V, W being non empty VectSpStr over K for v being Vector of V holds FunctionalFAF ((NulForm (V,W)),v) = 0Functional W
t11_bilinear:: for K being non empty ZeroStr for V, W being non empty VectSpStr over K for w being Vector of W holds FunctionalSAF ((NulForm (V,W)),w) = 0Functional V
t12_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, g being Form of V,W for w being Vector of W holds FunctionalSAF ((f + g),w) = (FunctionalSAF (f,w)) + (FunctionalSAF (g,w))
t13_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, g being Form of V,W for v being Vector of V holds FunctionalFAF ((f + g),v) = (FunctionalFAF (f,v)) + (FunctionalFAF (g,v))
t14_bilinear:: for K being non empty doubleLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W for a being Element of K for w being Vector of W holds FunctionalSAF ((a * f),w) = a * (FunctionalSAF (f,w))
t15_bilinear:: for K being non empty doubleLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W for a being Element of K for v being Vector of V holds FunctionalFAF ((a * f),v) = a * (FunctionalFAF (f,v))
t16_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W for w being Vector of W holds FunctionalSAF ((- f),w) = - (FunctionalSAF (f,w))
t17_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W for v being Vector of V holds FunctionalFAF ((- f),v) = - (FunctionalFAF (f,v))
t18_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, g being Form of V,W for w being Vector of W holds FunctionalSAF ((f - g),w) = (FunctionalSAF (f,w)) - (FunctionalSAF (g,w))
t19_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for f, g being Form of V,W for v being Vector of V holds FunctionalFAF ((f - g),v) = (FunctionalFAF (f,v)) - (FunctionalFAF (g,v))
t2_bilinear:: for K being non empty add-associative addLoopStr for V, W being non empty VectSpStr over K for f, g, h being Form of V,W holds (f + g) + h = f + (g + h)
t20_bilinear:: for K being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being Functional of V for v being Vector of V for w being Vector of W holds (FormFunctional (f,(0Functional W))) . (v,w) = 0. K
t21_bilinear:: for K being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr for V, W being non empty VectSpStr over K for g being Functional of W for v being Vector of V for w being Vector of W holds (FormFunctional ((0Functional V),g)) . (v,w) = 0. K
t22_bilinear:: for K being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being Functional of V holds FormFunctional (f,(0Functional W)) = NulForm (V,W)
t23_bilinear:: for K being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr for V, W being non empty VectSpStr over K for g being Functional of W holds FormFunctional ((0Functional V),g) = NulForm (V,W)
t24_bilinear:: for K being non empty multMagma for V, W being non empty VectSpStr over K for f being Functional of V for g being Functional of W for v being Vector of V holds FunctionalFAF ((FormFunctional (f,g)),v) = (f . v) * g
t25_bilinear:: for K being non empty commutative multMagma for V, W being non empty VectSpStr over K for f being Functional of V for g being Functional of W for w being Vector of W holds FunctionalSAF ((FormFunctional (f,g)),w) = (g . w) * f
t26_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for v, u being Vector of V for w being Vector of W for f being Form of V,W st f is additiveSAF holds f . ((v + u),w) = (f . (v,w)) + (f . (u,w))
t27_bilinear:: for K being non empty addLoopStr for V, W being non empty VectSpStr over K for v being Vector of V for u, w being Vector of W for f being Form of V,W st f is additiveFAF holds f . (v,(u + w)) = (f . (v,u)) + (f . (v,w))
t28_bilinear:: for K being non empty right_zeroed addLoopStr for V, W being non empty VectSpStr over K for v, u being Vector of V for w, t being Vector of W for f being additiveFAF additiveSAF Form of V,W holds f . ((v + u),(w + t)) = ((f . (v,w)) + (f . (v,t))) + ((f . (u,w)) + (f . (u,t)))
t29_bilinear:: for K being non empty right_complementable add-associative right_zeroed doubleLoopStr for V, W being non empty right_zeroed VectSpStr over K for f being additiveFAF Form of V,W for v being Vector of V holds f . (v,(0. W)) = 0. K
t3_bilinear:: for K being non empty right_complementable add-associative right_zeroed addLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W holds f - f = NulForm (V,W)
t30_bilinear:: for K being non empty right_complementable add-associative right_zeroed doubleLoopStr for V, W being non empty right_zeroed VectSpStr over K for f being additiveSAF Form of V,W for w being Vector of W holds f . ((0. V),w) = 0. K
t31_bilinear:: for K being non empty multMagma for V, W being non empty VectSpStr over K for v being Vector of V for w being Vector of W for a being Element of K for f being Form of V,W st f is homogeneousSAF holds f . ((a * v),w) = a * (f . (v,w))
t32_bilinear:: for K being non empty multMagma for V, W being non empty VectSpStr over K for v being Vector of V for w being Vector of W for a being Element of K for f being Form of V,W st f is homogeneousFAF holds f . (v,(a * w)) = a * (f . (v,w))
t33_bilinear:: for K being non empty right_complementable add-associative right_zeroed associative distributive left_unital doubleLoopStr for V, W being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over K for f being homogeneousSAF Form of V,W for w being Vector of W holds f . ((0. V),w) = 0. K
t34_bilinear:: for K being non empty right_complementable add-associative right_zeroed associative distributive left_unital doubleLoopStr for V, W being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over K for f being homogeneousFAF Form of V,W for v being Vector of V holds f . (v,(0. W)) = 0. K
t35_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for v, u being Vector of V for w being Vector of W for f being additiveSAF homogeneousSAF Form of V,W holds f . ((v - u),w) = (f . (v,w)) - (f . (u,w))
t36_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for v being Vector of V for w, t being Vector of W for f being additiveFAF homogeneousFAF Form of V,W holds f . (v,(w - t)) = (f . (v,w)) - (f . (v,t))
t37_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for v, u being Vector of V for w, t being Vector of W for f being bilinear-Form of V,W holds f . ((v - u),(w - t)) = ((f . (v,w)) - (f . (v,t))) - ((f . (u,w)) - (f . (u,t)))
t38_bilinear:: for K being non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital VectSpStr over K for v, u being Vector of V for w, t being Vector of W for a, b being Element of K for f being bilinear-Form of V,W holds f . ((v + (a * u)),(w + (b * t))) = ((f . (v,w)) + (b * (f . (v,t)))) + ((a * (f . (u,w))) + (a * (b * (f . (u,t)))))
t39_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for v, u being Vector of V for w, t being Vector of W for a, b being Element of K for f being bilinear-Form of V,W holds f . ((v - (a * u)),(w - (b * t))) = ((f . (v,w)) - (b * (f . (v,t)))) - ((a * (f . (u,w))) - (a * (b * (f . (u,t)))))
t4_bilinear:: for K being non empty right-distributive doubleLoopStr for V, W being non empty VectSpStr over K for a being Element of K for f, g being Form of V,W holds a * (f + g) = (a * f) + (a * g)
t40_bilinear:: for K being non empty right_complementable add-associative right_zeroed doubleLoopStr for V, W being non empty right_zeroed VectSpStr over K for f being Form of V,W st ( f is additiveFAF or f is additiveSAF ) holds ( f is constant iff for v being Vector of V for w being Vector of W holds f . (v,w) = 0. K )
t41_bilinear:: for K being ZeroStr for V being non empty VectSpStr over K for f being Form of V,V holds ( leftker f c= diagker f & rightker f c= diagker f )
t42_bilinear:: for K being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being additiveSAF homogeneousSAF Form of V,W holds leftker f is linearly-closed
t43_bilinear:: for K being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being additiveFAF homogeneousFAF Form of V,W holds rightker f is linearly-closed
t44_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V being VectSp of K for W being non empty VectSpStr over K for f being additiveSAF homogeneousSAF Form of V,W holds rightker f = rightker (LQForm f)
t45_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V being non empty VectSpStr over K for W being VectSp of K for f being additiveFAF homogeneousFAF Form of V,W holds leftker f = leftker (RQForm f)
t46_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for f being bilinear-Form of V,W holds RKer f = RKer (LQForm f)
t47_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for f being bilinear-Form of V,W holds LKer f = LKer (RQForm f)
t48_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for f being bilinear-Form of V,W holds ( QForm f = RQForm (LQForm f) & QForm f = LQForm (RQForm f) )
t49_bilinear:: for K being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr for V, W being VectSp of K for f being bilinear-Form of V,W holds ( leftker (QForm f) = leftker (RQForm (LQForm f)) & rightker (QForm f) = rightker (RQForm (LQForm f)) & leftker (QForm f) = leftker (LQForm (RQForm f)) & rightker (QForm f) = rightker (LQForm (RQForm f)) )
t5_bilinear:: for K being non empty left-distributive doubleLoopStr for V, W being non empty VectSpStr over K for a, b being Element of K for f being Form of V,W holds (a + b) * f = (a * f) + (b * f)
t50_bilinear:: for K being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being Functional of V for g being Functional of W holds ker f c= leftker (FormFunctional (f,g))
t51_bilinear:: for K being non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being Functional of V for g being Functional of W st g <> 0Functional W holds leftker (FormFunctional (f,g)) = ker f
t52_bilinear:: for K being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being Functional of V for g being Functional of W holds ker g c= rightker (FormFunctional (f,g))
t53_bilinear:: for K being non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr for V, W being non empty VectSpStr over K for f being Functional of V for g being Functional of W st f <> 0Functional V holds rightker (FormFunctional (f,g)) = ker g
t54_bilinear:: for K being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr for V being VectSp of K for W being non empty VectSpStr over K for f being linear-Functional of V for g being Functional of W st g <> 0Functional W holds ( LKer (FormFunctional (f,g)) = Ker f & LQForm (FormFunctional (f,g)) = FormFunctional ((CQFunctional f),g) )
t55_bilinear:: for K being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr for V being non empty VectSpStr over K for W being VectSp of K for f being Functional of V for g being linear-Functional of W st f <> 0Functional V holds ( RKer (FormFunctional (f,g)) = Ker g & RQForm (FormFunctional (f,g)) = FormFunctional (f,(CQFunctional g)) )
t56_bilinear:: for K being Field for V, W being non trivial VectSp of K for f being V17() linear-Functional of V for g being V17() linear-Functional of W holds QForm (FormFunctional (f,g)) = FormFunctional ((CQFunctional f),(CQFunctional g))
t57_bilinear:: for K being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr for V being non empty VectSpStr over K for f being symmetric bilinear-Form of V,V holds leftker f = rightker f
t58_bilinear:: for K being non empty right_complementable add-associative right_zeroed addLoopStr for V being non empty VectSpStr over K for f being additiveFAF additiveSAF alternating Form of V,V for v, w being Vector of V holds f . (v,w) = - (f . (w,v))
t59_bilinear:: for K being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr for V being non empty VectSpStr over K for f being alternating bilinear-Form of V,V holds leftker f = rightker f
t6_bilinear:: for K being non empty associative doubleLoopStr for V, W being non empty VectSpStr over K for a, b being Element of K for f being Form of V,W holds (a * b) * f = a * (b * f)
t7_bilinear:: for K being non empty left_unital doubleLoopStr for V, W being non empty VectSpStr over K for f being Form of V,W holds (1. K) * f = f
t8_bilinear:: for K being non empty 1-sorted for V, W being non empty VectSpStr over K for f being Form of V,W for v being Vector of V holds ( dom (FunctionalFAF (f,v)) = the carrier of W & rng (FunctionalFAF (f,v)) c= the carrier of K & ( for w being Vector of W holds (FunctionalFAF (f,v)) . w = f . (v,w) ) )
t9_bilinear:: for K being non empty 1-sorted for V, W being non empty VectSpStr over K for f being Form of V,W for w being Vector of W holds ( dom (FunctionalSAF (f,w)) = the carrier of V & rng (FunctionalSAF (f,w)) c= the carrier of K & ( for v being Vector of V holds (FunctionalSAF (f,w)) . v = f . (v,w) ) )
d1_binari_2:: for n being Nat for b2 being Tuple of n, BOOLEAN holds ( b2 = Bin1 n iff for i being Nat st i in Seg n holds b2 /. i = IFEQ (i,1,TRUE,FALSE) );
d2_binari_2:: for n being non empty Nat for x being Tuple of n, BOOLEAN holds Neg2 x = ('not' x) + (Bin1 n);
d3_binari_2:: for n being Nat for x being Tuple of n, BOOLEAN holds ( ( x /. n = FALSE implies Intval x = Absval x ) & ( not x /. n = FALSE implies Intval x = (Absval x) - (2 to_power n) ) );
d4_binari_2:: for n being non empty Nat for z1, z2 being Tuple of n, BOOLEAN holds Int_add_ovfl (z1,z2) = (('not' (z1 /. n)) '&' ('not' (z2 /. n))) '&' ((carry (z1,z2)) /. n);
d5_binari_2:: for n being non empty Nat for z1, z2 being Tuple of n, BOOLEAN holds Int_add_udfl (z1,z2) = ((z1 /. n) '&' (z2 /. n)) '&' ('not' ((carry (z1,z2)) /. n));
d6_binari_2:: for n being non empty Nat for x, y, b4 being Tuple of n, BOOLEAN holds ( b4 = x - y iff for i being Nat st i in Seg n holds b4 /. i = ((x /. i) 'xor' ((Neg2 y) /. i)) 'xor' ((carry (x,(Neg2 y))) /. i) );
t1_binari_2:: for z1 being Tuple of 2, BOOLEAN st z1 = <*FALSE*> ^ <*FALSE*> holds Intval z1 = 0
t10_binari_2:: for m being non empty Nat for z being Tuple of m, BOOLEAN for d being Element of BOOLEAN holds Intval (z ^ <*d*>) = (Absval z) - (IFEQ (d,FALSE,0,(2 to_power m)))
t11_binari_2:: for m being non empty Nat for z1, z2 being Tuple of m, BOOLEAN for d1, d2 being Element of BOOLEAN holds ((Intval ((z1 ^ <*d1*>) + (z2 ^ <*d2*>))) + (IFEQ ((Int_add_ovfl ((z1 ^ <*d1*>),(z2 ^ <*d2*>))),FALSE,0,(2 to_power (m + 1))))) - (IFEQ ((Int_add_udfl ((z1 ^ <*d1*>),(z2 ^ <*d2*>))),FALSE,0,(2 to_power (m + 1)))) = (Intval (z1 ^ <*d1*>)) + (Intval (z2 ^ <*d2*>))
t12_binari_2:: for m being non empty Nat for z1, z2 being Tuple of m, BOOLEAN for d1, d2 being Element of BOOLEAN holds Intval ((z1 ^ <*d1*>) + (z2 ^ <*d2*>)) = (((Intval (z1 ^ <*d1*>)) + (Intval (z2 ^ <*d2*>))) - (IFEQ ((Int_add_ovfl ((z1 ^ <*d1*>),(z2 ^ <*d2*>))),FALSE,0,(2 to_power (m + 1))))) + (IFEQ ((Int_add_udfl ((z1 ^ <*d1*>),(z2 ^ <*d2*>))),FALSE,0,(2 to_power (m + 1))))
t13_binari_2:: for m being non empty Nat for x being Tuple of m, BOOLEAN holds Absval ('not' x) = ((- (Absval x)) + (2 to_power m)) - 1
t14_binari_2:: for m being non empty Nat for z being Tuple of m, BOOLEAN for d being Element of BOOLEAN holds Neg2 (z ^ <*d*>) = (Neg2 z) ^ <*(('not' d) 'xor' (add_ovfl (('not' z),(Bin1 m))))*>
t15_binari_2:: for m being non empty Nat for z being Tuple of m, BOOLEAN for d being Element of BOOLEAN holds (Intval (Neg2 (z ^ <*d*>))) + (IFEQ ((Int_add_ovfl (('not' (z ^ <*d*>)),(Bin1 (m + 1)))),FALSE,0,(2 to_power (m + 1)))) = - (Intval (z ^ <*d*>))
t16_binari_2:: for m being non empty Nat for z being Tuple of m, BOOLEAN for d being Element of BOOLEAN holds Neg2 (Neg2 (z ^ <*d*>)) = z ^ <*d*>
t17_binari_2:: for m being non empty Nat for x, y being Tuple of m, BOOLEAN holds x - y = x + (Neg2 y)
t18_binari_2:: for m being non empty Nat for z1, z2 being Tuple of m, BOOLEAN for d1, d2 being Element of BOOLEAN holds (z1 ^ <*d1*>) - (z2 ^ <*d2*>) = (z1 + (Neg2 z2)) ^ <*(((d1 'xor' ('not' d2)) 'xor' (add_ovfl (('not' z2),(Bin1 m)))) 'xor' (add_ovfl (z1,(Neg2 z2))))*>
t19_binari_2:: for m being non empty Nat for z1, z2 being Tuple of m, BOOLEAN for d1, d2 being Element of BOOLEAN holds (((Intval ((z1 ^ <*d1*>) - (z2 ^ <*d2*>))) + (IFEQ ((Int_add_ovfl ((z1 ^ <*d1*>),(Neg2 (z2 ^ <*d2*>)))),FALSE,0,(2 to_power (m + 1))))) - (IFEQ ((Int_add_udfl ((z1 ^ <*d1*>),(Neg2 (z2 ^ <*d2*>)))),FALSE,0,(2 to_power (m + 1))))) + (IFEQ ((Int_add_ovfl (('not' (z2 ^ <*d2*>)),(Bin1 (m + 1)))),FALSE,0,(2 to_power (m + 1)))) = (Intval (z1 ^ <*d1*>)) - (Intval (z2 ^ <*d2*>))
t2_binari_2:: for z1 being Tuple of 2, BOOLEAN st z1 = <*TRUE*> ^ <*FALSE*> holds Intval z1 = 1
t3_binari_2:: for z1 being Tuple of 2, BOOLEAN st z1 = <*FALSE*> ^ <*TRUE*> holds Intval z1 = - 2
t4_binari_2:: for z1 being Tuple of 2, BOOLEAN st z1 = <*TRUE*> ^ <*TRUE*> holds Intval z1 = - 1
t5_binari_2:: for n, i being Nat st i in Seg n & i = 1 holds (Bin1 n) /. i = TRUE
t6_binari_2:: for n, i being Nat st i in Seg n & i <> 1 holds (Bin1 n) /. i = FALSE
t7_binari_2:: for m being non empty Nat holds Bin1 (m + 1) = (Bin1 m) ^ <*FALSE*>
t8_binari_2:: for m being non empty Nat holds Intval ((Bin1 m) ^ <*FALSE*>) = 1
t9_binari_2:: for m being non empty Nat for z being Tuple of m, BOOLEAN for d being Element of BOOLEAN holds 'not' (z ^ <*d*>) = ('not' z) ^ <*('not' d)*>
d1_binari_3:: for n, k being Nat for b3 being Tuple of n, BOOLEAN holds ( b3 = n -BinarySequence k iff for i being Nat st i in Seg n holds b3 /. i = IFEQ (((k div (2 to_power (i -' 1))) mod 2),0,FALSE,TRUE) );
t1_binari_3:: for n being non empty Nat for F being Tuple of n, BOOLEAN holds Absval F < 2 to_power n
t10_binari_3:: Bin1 1 = <*TRUE*>
t11_binari_3:: for n being non empty Nat holds Absval (Bin1 n) = 1
t12_binari_3:: for x, y being Element of BOOLEAN holds ( ( not x 'or' y = TRUE or x = TRUE or y = TRUE ) & ( ( x = TRUE or y = TRUE ) implies x 'or' y = TRUE ) & ( x 'or' y = FALSE implies ( x = FALSE & y = FALSE ) ) & ( x = FALSE & y = FALSE implies x 'or' y = FALSE ) )
t13_binari_3:: for x, y being Element of BOOLEAN holds ( add_ovfl (<*x*>,<*y*>) = TRUE iff ( x = TRUE & y = TRUE ) )
t14_binari_3:: 'not' <*FALSE*> = <*TRUE*>
t15_binari_3:: 'not' <*TRUE*> = <*FALSE*>
t16_binari_3:: <*FALSE*> + <*FALSE*> = <*FALSE*>
t17_binari_3:: ( <*FALSE*> + <*TRUE*> = <*TRUE*> & <*TRUE*> + <*FALSE*> = <*TRUE*> )
t18_binari_3:: <*TRUE*> + <*TRUE*> = <*FALSE*>
t19_binari_3:: for n being non empty Nat for x, y being Tuple of n, BOOLEAN st x /. n = TRUE & (carry (x,(Bin1 n))) /. n = TRUE holds for k being non empty Nat st k <> 1 & k <= n holds ( x /. k = TRUE & (carry (x,(Bin1 n))) /. k = TRUE )
t2_binari_3:: for n being non empty Nat for F1, F2 being Tuple of n, BOOLEAN st Absval F1 = Absval F2 holds F1 = F2
t20_binari_3:: for n being non empty Nat for x being Tuple of n, BOOLEAN st x /. n = TRUE & (carry (x,(Bin1 n))) /. n = TRUE holds carry (x,(Bin1 n)) = 'not' (Bin1 n)
t21_binari_3:: for n being non empty Nat for x, y being Tuple of n, BOOLEAN st y = 0* n & x /. n = TRUE & (carry (x,(Bin1 n))) /. n = TRUE holds x = 'not' y
t22_binari_3:: for n being non empty Nat for y being Tuple of n, BOOLEAN st y = 0* n holds carry (('not' y),(Bin1 n)) = 'not' (Bin1 n)
t23_binari_3:: for n being non empty Nat for x, y being Tuple of n, BOOLEAN st y = 0* n holds ( add_ovfl (x,(Bin1 n)) = TRUE iff x = 'not' y )
t24_binari_3:: for n being non empty Nat for z being Tuple of n, BOOLEAN st z = 0* n holds ('not' z) + (Bin1 n) = z
t25_binari_3:: for n being Nat holds n -BinarySequence 0 = 0* n
t26_binari_3:: for n, k being Nat st k < 2 to_power n holds ((n + 1) -BinarySequence k) . (n + 1) = FALSE
t27_binari_3:: for n being non empty Nat for k being Nat st k < 2 to_power n holds (n + 1) -BinarySequence k = (n -BinarySequence k) ^ <*FALSE*>
t28_binari_3:: for i being Nat holds (i + 1) -BinarySequence (2 to_power i) = (0* i) ^ <*1*>
t29_binari_3:: for n being non empty Nat for k being Nat st 2 to_power n <= k & k < 2 to_power (n + 1) holds ((n + 1) -BinarySequence k) . (n + 1) = TRUE by Lm2;
t3_binari_3:: for t1, t2 being FinSequence st Rev t1 = Rev t2 holds t1 = t2
t30_binari_3:: for n being non empty Nat for k being Nat st 2 to_power n <= k & k < 2 to_power (n + 1) holds (n + 1) -BinarySequence k = (n -BinarySequence (k -' (2 to_power n))) ^ <*TRUE*> by Lm3;
t31_binari_3:: for n being non empty Nat for k being Nat st k < 2 to_power n holds for x being Tuple of n, BOOLEAN st x = 0* n holds ( n -BinarySequence k = 'not' x iff k = (2 to_power n) - 1 ) by Lm4;
t32_binari_3:: for n being non empty Nat for k being Nat st k + 1 < 2 to_power n holds add_ovfl ((n -BinarySequence k),(Bin1 n)) = FALSE
t33_binari_3:: for n being non empty Nat for k being Nat st k + 1 < 2 to_power n holds n -BinarySequence (k + 1) = (n -BinarySequence k) + (Bin1 n)
t34_binari_3:: for n, i being Nat holds (n + 1) -BinarySequence i = <*(i mod 2)*> ^ (n -BinarySequence (i div 2))
t35_binari_3:: for n being non empty Nat for k being Nat st k < 2 to_power n holds Absval (n -BinarySequence k) = k
t36_binari_3:: for n being non empty Nat for x being Tuple of n, BOOLEAN holds n -BinarySequence (Absval x) = x
t4_binari_3:: for n being Nat holds 0* n in BOOLEAN *
t5_binari_3:: for n being Nat for y being Tuple of n, BOOLEAN st y = 0* n holds 'not' y = n |-> 1
t6_binari_3:: for n being non empty Nat for F being Tuple of n, BOOLEAN st F = 0* n holds Absval F = 0
t7_binari_3:: for n being non empty Nat for F being Tuple of n, BOOLEAN st F = 0* n holds Absval ('not' F) = (2 to_power n) - 1
t8_binari_3:: for n being Nat holds Rev (0* n) = 0* n
t9_binari_3:: for n being Nat for y being Tuple of n, BOOLEAN st y = 0* n holds Rev ('not' y) = 'not' y
d1_binari_4:: for m, j, b3 being Nat holds ( b3 = MajP (m,j) iff ( 2 to_power b3 >= j & b3 >= m & ( for k being Nat st 2 to_power k >= j & k >= m holds k >= b3 ) ) );
d2_binari_4:: for m being Nat for i being Integer holds ( ( i < 0 implies 2sComplement (m,i) = m -BinarySequence (abs ((2 to_power (MajP (m,(abs i)))) + i)) ) & ( not i < 0 implies 2sComplement (m,i) = m -BinarySequence (abs i) ) );
t1_binari_4:: for m being Nat st m > 0 holds m * 2 >= m + 1
t10_binari_4:: for n being non empty Nat for l, m being Nat st l + m <= (2 to_power n) - 1 holds add_ovfl ((n -BinarySequence l),(n -BinarySequence m)) = FALSE
t11_binari_4:: for n being non empty Nat for l, m being Nat st l + m <= (2 to_power n) - 1 holds Absval ((n -BinarySequence l) + (n -BinarySequence m)) = l + m
t12_binari_4:: for n being non empty Nat for z being Tuple of n, BOOLEAN st z /. n = TRUE holds Absval z >= 2 to_power (n -' 1)
t13_binari_4:: for n being non empty Nat for l, m being Nat st l + m <= (2 to_power (n -' 1)) - 1 holds (carry ((n -BinarySequence l),(n -BinarySequence m))) /. n = FALSE
t14_binari_4:: for l, m being Nat for n being non empty Nat st l + m <= (2 to_power (n -' 1)) - 1 holds Intval ((n -BinarySequence l) + (n -BinarySequence m)) = l + m
t15_binari_4:: for z being Tuple of 1, BOOLEAN st z = <*TRUE*> holds Intval z = - 1
t16_binari_4:: for z being Tuple of 1, BOOLEAN st z = <*FALSE*> holds Intval z = 0
t17_binari_4:: for x being boolean set holds TRUE 'or' x = TRUE ;
t18_binari_4:: for n being non empty Nat holds ( 0 <= (2 to_power (n -' 1)) - 1 & - (2 to_power (n -' 1)) <= 0 )
t19_binari_4:: for n being non empty Nat for x, y being Tuple of n, BOOLEAN st x = 0* n & y = 0* n holds x,y are_summable
t2_binari_4:: for m being Nat holds 2 to_power m >= m
t20_binari_4:: for n being non empty Nat for i being Integer holds (i * n) mod n = 0
t21_binari_4:: for j, k, m being Nat st j >= k holds MajP (m,j) >= MajP (m,k)
t22_binari_4:: for l, m, j being Nat st l >= m holds MajP (l,j) >= MajP (m,j)
t23_binari_4:: for m being Nat st m >= 1 holds MajP (m,1) = m
t24_binari_4:: for j, m being Nat st j <= 2 to_power m holds MajP (m,j) = m
t25_binari_4:: for j, m being Nat st j > 2 to_power m holds MajP (m,j) > m
t26_binari_4:: for m being Nat holds 2sComplement (m,0) = 0* m
t27_binari_4:: for n being non empty Nat for i being Integer st i <= (2 to_power (n -' 1)) - 1 & - (2 to_power (n -' 1)) <= i holds Intval (2sComplement (n,i)) = i
t28_binari_4:: for n being non empty Nat for h, i being Integer st ( ( h >= 0 & i >= 0 ) or ( h < 0 & i < 0 ) ) & h mod (2 to_power n) = i mod (2 to_power n) holds 2sComplement (n,h) = 2sComplement (n,i) by Lm5, Lm6;
t29_binari_4:: for n being non empty Nat for h, i being Integer st ( ( h >= 0 & i >= 0 ) or ( h < 0 & i < 0 ) ) & h,i are_congruent_mod 2 to_power n holds 2sComplement (n,h) = 2sComplement (n,i)
t3_binari_4:: for m being Nat holds (0* m) + (0* m) = 0* m
t30_binari_4:: for n being non empty Nat for l, m being Nat st l mod (2 to_power n) = m mod (2 to_power n) holds n -BinarySequence l = n -BinarySequence m
t31_binari_4:: for n being non empty Nat for l, m being Nat st l,m are_congruent_mod 2 to_power n holds n -BinarySequence l = n -BinarySequence m
t32_binari_4:: for n being non empty Nat for i being Integer for j being Nat st 1 <= j & j <= n holds (2sComplement ((n + 1),i)) /. j = (2sComplement (n,i)) /. j
t33_binari_4:: for m being Nat for i being Integer ex x being Element of BOOLEAN st 2sComplement ((m + 1),i) = (2sComplement (m,i)) ^ <*x*>
t34_binari_4:: for m, l being Nat ex x being Element of BOOLEAN st (m + 1) -BinarySequence l = (m -BinarySequence l) ^ <*x*>
t35_binari_4:: for h, i being Integer for n being non empty Nat st - (2 to_power n) <= h + i & h < 0 & i < 0 & - (2 to_power (n -' 1)) <= h & - (2 to_power (n -' 1)) <= i holds (carry ((2sComplement ((n + 1),h)),(2sComplement ((n + 1),i)))) /. (n + 1) = TRUE
t36_binari_4:: for h, i being Integer for n being non empty Nat st h + i <= (2 to_power (n -' 1)) - 1 & h >= 0 & i >= 0 holds Intval ((2sComplement (n,h)) + (2sComplement (n,i))) = h + i
t37_binari_4:: for h, i being Integer for n being non empty Nat st - (2 to_power ((n + 1) -' 1)) <= h + i & h < 0 & i < 0 & - (2 to_power (n -' 1)) <= h & - (2 to_power (n -' 1)) <= i holds Intval ((2sComplement ((n + 1),h)) + (2sComplement ((n + 1),i))) = h + i
t38_binari_4:: for h, i being Integer for n being non empty Nat st - (2 to_power (n -' 1)) <= h & h <= (2 to_power (n -' 1)) - 1 & - (2 to_power (n -' 1)) <= i & i <= (2 to_power (n -' 1)) - 1 & - (2 to_power (n -' 1)) <= h + i & h + i <= (2 to_power (n -' 1)) - 1 & ( ( h >= 0 & i < 0 ) or ( h < 0 & i >= 0 ) ) holds Intval ((2sComplement (n,h)) + (2sComplement (n,i))) = h + i
t4_binari_4:: for k, m, l being Nat st k <= l & l <= m & not k = l holds ( k + 1 <= l & l <= m )
t5_binari_4:: for n being non empty Nat for x, y being Tuple of n, BOOLEAN st x = 0* n & y = 0* n holds carry (x,y) = 0* n
t6_binari_4:: for n being non empty Nat for x, y being Tuple of n, BOOLEAN st x = 0* n & y = 0* n holds x + y = 0* n
t7_binari_4:: for n being non empty Nat for F being Tuple of n, BOOLEAN st F = 0* n holds Intval F = 0
t8_binari_4:: for l, m, k being Nat st l + m <= k - 1 holds ( l < k & m < k )
t9_binari_4:: for g, h, i being Integer st g <= h + i & h < 0 & i < 0 holds ( g < h & g < i )
d1_binarith:: for n being Nat for x, b3 being Tuple of n, BOOLEAN holds ( b3 = 'not' x iff for i being Nat st i in Seg n holds b3 /. i = 'not' (x /. i) );
d2_binarith:: for n being non empty Nat for x, y, b4 being Tuple of n, BOOLEAN holds ( b4 = carry (x,y) iff ( b4 /. 1 = FALSE & ( for i being Nat st 1 <= i & i < n holds b4 /. (i + 1) = (((x /. i) '&' (y /. i)) 'or' ((x /. i) '&' (b4 /. i))) 'or' ((y /. i) '&' (b4 /. i)) ) ) );
d3_binarith:: for n being Nat for x being Tuple of n, BOOLEAN for b3 being Tuple of n, NAT holds ( b3 = Binary x iff for i being Nat st i in Seg n holds b3 /. i = IFEQ ((x /. i),FALSE,0,(2 to_power (i -' 1))) );
d4_binarith:: for n being Nat for x being Tuple of n, BOOLEAN holds Absval x = addnat $$ (Binary x);
d5_binarith:: for n being non zero Nat for x, y, b4 being Tuple of n, BOOLEAN holds ( b4 = x + y iff for i being Nat st i in Seg n holds b4 /. i = ((x /. i) 'xor' (y /. i)) 'xor' ((carry (x,y)) /. i) );
d6_binarith:: for n being non zero Nat for z1, z2 being Tuple of n, BOOLEAN holds add_ovfl (z1,z2) = (((z1 /. n) '&' (z2 /. n)) 'or' ((z1 /. n) '&' ((carry (z1,z2)) /. n))) 'or' ((z2 /. n) '&' ((carry (z1,z2)) /. n));
d7_binarith:: for n being non zero Nat for z1, z2 being Tuple of n, BOOLEAN holds ( z1,z2 are_summable iff add_ovfl (z1,z2) = FALSE );
t1_binarith:: for i, n being Nat for D being non empty set for d being Element of D for z being Tuple of n,D st i in Seg n holds (z ^ <*d*>) /. i = z /. i
t10_binarith:: for x being boolean set holds x 'or' TRUE = TRUE ;
t11_binarith:: for x, y, z being boolean set holds (x 'or' y) 'or' z = x 'or' (y 'or' z) ;
t12_binarith:: for x being boolean set holds x 'or' x = x ;
t13_binarith:: TRUE 'xor' FALSE = TRUE ;
t14_binarith:: for z1 being Tuple of 1, BOOLEAN holds ( z1 = <*FALSE*> or z1 = <*TRUE*> )
t15_binarith:: for z1 being Tuple of 1, BOOLEAN st z1 = <*FALSE*> holds Absval z1 = 0
t16_binarith:: for z1 being Tuple of 1, BOOLEAN st z1 = <*TRUE*> holds Absval z1 = 1
t17_binarith:: for n being non zero Nat for z1, z2 being Tuple of n, BOOLEAN for d1, d2 being Element of BOOLEAN for i being Nat st i in Seg n holds (carry ((z1 ^ <*d1*>),(z2 ^ <*d2*>))) /. i = (carry (z1,z2)) /. i
t18_binarith:: for n being non zero Nat for z1, z2 being Tuple of n, BOOLEAN for d1, d2 being Element of BOOLEAN holds add_ovfl (z1,z2) = (carry ((z1 ^ <*d1*>),(z2 ^ <*d2*>))) /. (n + 1)
t19_binarith:: for n being non zero Nat for z1, z2 being Tuple of n, BOOLEAN for d1, d2 being Element of BOOLEAN holds (z1 ^ <*d1*>) + (z2 ^ <*d2*>) = (z1 + z2) ^ <*((d1 'xor' d2) 'xor' (add_ovfl (z1,z2)))*>
t2_binarith:: for n being Nat for D being non empty set for d being Element of D for z being Tuple of n,D holds (z ^ <*d*>) /. (n + 1) = d
t20_binarith:: for n being non zero Nat for z being Tuple of n, BOOLEAN for d being Element of BOOLEAN holds Absval (z ^ <*d*>) = (Absval z) + (IFEQ (d,FALSE,0,(2 to_power n)))
t21_binarith:: for n being non zero Nat for z1, z2 being Tuple of n, BOOLEAN holds (Absval (z1 + z2)) + (IFEQ ((add_ovfl (z1,z2)),FALSE,0,(2 to_power n))) = (Absval z1) + (Absval z2)
t22_binarith:: for n being non zero Nat for z1, z2 being Tuple of n, BOOLEAN st z1,z2 are_summable holds Absval (z1 + z2) = (Absval z1) + (Absval z2)
t3_binarith:: for x being boolean set holds x 'or' FALSE = x ;
t4_binarith:: for x, y being boolean set holds 'not' (x '&' y) = ('not' x) 'or' ('not' y) ;
t5_binarith:: for x, y being boolean set holds 'not' (x 'or' y) = ('not' x) '&' ('not' y) ;
t6_binarith:: for x, y being boolean set holds x '&' y = 'not' (('not' x) 'or' ('not' y)) ;
t7_binarith:: for x being boolean set holds TRUE 'xor' x = 'not' x ;
t8_binarith:: for x being boolean set holds FALSE 'xor' x = x ;
t9_binarith:: for x being boolean set holds x '&' x = x ;
d1_binom:: for R being non empty addLoopStr for p, q, b4 being FinSequence of the carrier of R holds ( b4 = p + q iff ( dom b4 = dom p & ( for i being Nat st 1 <= i & i <= len b4 holds b4 /. i = (p /. i) + (q /. i) ) ) );
d2_binom:: for R being non empty unital multMagma for a being Element of R for n being Nat holds a |^ n = (power R) . (a,n);
d3_binom:: for R being non empty addLoopStr for b2 being Function of [:NAT, the carrier of R:], the carrier of R holds ( b2 = Nat-mult-left R iff for a being Element of R holds ( b2 . (0,a) = 0. R & ( for n being Element of NAT holds b2 . ((n + 1),a) = a + (b2 . (n,a)) ) ) );
d4_binom:: for R being non empty addLoopStr for b2 being Function of [: the carrier of R,NAT:], the carrier of R holds ( b2 = Nat-mult-right R iff for a being Element of R holds ( b2 . (a,0) = 0. R & ( for n being Element of NAT holds b2 . (a,(n + 1)) = (b2 . (a,n)) + a ) ) );
d5_binom:: for R being non empty addLoopStr for a being Element of R for n being Element of NAT holds n * a = (Nat-mult-left R) . (n,a);
d6_binom:: for R being non empty addLoopStr for a being Element of R for n being Element of NAT holds a * n = (Nat-mult-right R) . (a,n);
d7_binom:: for R being non empty unital doubleLoopStr for a, b being Element of R for n being Element of NAT for b5 being FinSequence of the carrier of R holds ( b5 = (a,b) In_Power n iff ( len b5 = n + 1 & ( for i, l, m being Element of NAT st i in dom b5 & m = i - 1 & l = n - m holds b5 /. i = ((n choose m) * (a |^ l)) * (b |^ m) ) ) );
t1_binom:: for R being non empty left_add-cancelable left-distributive right_zeroed doubleLoopStr for a being Element of R holds (0. R) * a = 0. R
t10_binom:: for R being non empty unital associative multMagma for a being Element of R for n, m being Nat holds a |^ (n + m) = (a |^ n) * (a |^ m)
t11_binom:: for R being non empty unital associative multMagma for a being Element of R for n, m being Nat holds (a |^ n) |^ m = a |^ (n * m)
t12_binom:: for R being non empty addLoopStr for a being Element of R holds ( 0 * a = 0. R & a * 0 = 0. R ) by Def3, Def4;
t13_binom:: for R being non empty right_zeroed addLoopStr for a being Element of R holds 1 * a = a
t14_binom:: for R being non empty left_zeroed addLoopStr for a being Element of R holds a * 1 = a
t15_binom:: for R being non empty left_zeroed add-associative addLoopStr for a being Element of R for n, m being Element of NAT holds (n + m) * a = (n * a) + (m * a)
t16_binom:: for R being non empty add-associative right_zeroed addLoopStr for a being Element of R for n, m being Element of NAT holds a * (n + m) = (a * n) + (a * m)
t17_binom:: for R being non empty left_zeroed add-associative right_zeroed addLoopStr for a being Element of R for n being Element of NAT holds n * a = a * n
t18_binom:: for R being non empty Abelian addLoopStr for a being Element of R for n being Element of NAT holds n * a = a * n
t19_binom:: for R being non empty left_add-cancelable left_zeroed left-distributive add-associative right_zeroed doubleLoopStr for a, b being Element of R for n being Element of NAT holds (n * a) * b = n * (a * b)
t2_binom:: for R being non empty right_add-cancelable left_zeroed right-distributive doubleLoopStr for a being Element of R holds a * (0. R) = 0. R
t20_binom:: for R being non empty right_add-cancelable left_zeroed distributive add-associative right_zeroed doubleLoopStr for a, b being Element of R for n being Element of NAT holds b * (n * a) = (b * a) * n
t21_binom:: for R being non empty add-cancelable left_zeroed distributive add-associative right_zeroed doubleLoopStr for a, b being Element of R for n being Element of NAT holds (a * n) * b = a * (n * b)
t22_binom:: for R being non empty unital right_zeroed doubleLoopStr for a, b being Element of R holds (a,b) In_Power 0 = <*(1_ R)*>
t23_binom:: for R being non empty unital right_zeroed doubleLoopStr for a, b being Element of R for n being Element of NAT holds ((a,b) In_Power n) . 1 = a |^ n
t24_binom:: for R being non empty unital right_zeroed doubleLoopStr for a, b being Element of R for n being Element of NAT holds ((a,b) In_Power n) . (n + 1) = b |^ n
t25_binom:: for R being non empty add-cancelable left_zeroed unital associative commutative distributive Abelian add-associative right_zeroed doubleLoopStr for a, b being Element of R for n being Element of NAT holds (a + b) |^ n = Sum ((a,b) In_Power n)
t26_binom:: for C, D being non empty set for b being Element of D for F being Function of [:C,D:],D ex g being Function of [:NAT,C:],D st for a being Element of C holds ( g . (0,a) = b & ( for n being Element of NAT holds g . ((n + 1),a) = F . (a,(g . (n,a))) ) ) by Lm1;
t27_binom:: for C, D being non empty set for b being Element of D for F being Function of [:D,C:],D ex g being Function of [:C,NAT:],D st for a being Element of C holds ( g . (a,0) = b & ( for n being Element of NAT holds g . (a,(n + 1)) = F . ((g . (a,n)),a) ) ) by Lm2;
t3_binom:: for L being non empty left_zeroed addLoopStr for a being Element of L holds Sum <*a*> = a
t4_binom:: for R being non empty right_add-cancelable left_zeroed right-distributive doubleLoopStr for a being Element of R for p being FinSequence of the carrier of R holds Sum (a * p) = a * (Sum p)
t5_binom:: for R being non empty left_add-cancelable left-distributive right_zeroed doubleLoopStr for a being Element of R for p being FinSequence of the carrier of R holds Sum (p * a) = (Sum p) * a
t6_binom:: for R being non empty commutative multMagma for a being Element of R for p being FinSequence of the carrier of R holds p * a = a * p
t7_binom:: for R being non empty Abelian add-associative right_zeroed addLoopStr for p, q being FinSequence of the carrier of R st dom p = dom q holds Sum (p + q) = (Sum p) + (Sum q)
t8_binom:: for R being non empty unital multMagma for a being Element of R holds ( a |^ 0 = 1_ R & a |^ 1 = a )
t9_binom:: for R being non empty unital associative commutative multMagma for a, b being Element of R for n being Nat holds (a * b) |^ n = (a |^ n) * (b |^ n)
d1_binop_1:: for f being Function for a, b being set holds f . (a,b) = f . [a,b];
d10_binop_1:: for A being set for o9, o being BinOp of A holds ( o9 is_right_distributive_wrt o iff for a, b, c being Element of A holds o9 . ((o . (a,b)),c) = o . ((o9 . (a,c)),(o9 . (b,c))) );
d11_binop_1:: for A being set for o9, o being BinOp of A holds ( o9 is_distributive_wrt o iff ( o9 is_left_distributive_wrt o & o9 is_right_distributive_wrt o ) );
d12_binop_1:: for A being set for u being UnOp of A for o being BinOp of A holds ( u is_distributive_wrt o iff for a, b being Element of A holds u . (o . (a,b)) = o . ((u . a),(u . b)) );
d13_binop_1:: canceled;
d14_binop_1:: canceled;
d15_binop_1:: canceled;
d16_binop_1:: for A being non empty set for e being Element of A for o being BinOp of A holds ( e is_a_left_unity_wrt o iff for a being Element of A holds o . (e,a) = a );
d17_binop_1:: for A being non empty set for e being Element of A for o being BinOp of A holds ( e is_a_right_unity_wrt o iff for a being Element of A holds o . (a,e) = a );
d18_binop_1:: for A being non empty set for o9, o being BinOp of A holds ( o9 is_left_distributive_wrt o iff for a, b, c being Element of A holds o9 . (a,(o . (b,c))) = o . ((o9 . (a,b)),(o9 . (a,c))) );
d19_binop_1:: for A being non empty set for o9, o being BinOp of A holds ( o9 is_right_distributive_wrt o iff for a, b, c being Element of A holds o9 . ((o . (a,b)),c) = o . ((o9 . (a,c)),(o9 . (b,c))) );
d2_binop_1:: for A being set for o being BinOp of A holds ( o is commutative iff for a, b being Element of A holds o . (a,b) = o . (b,a) );
d20_binop_1:: for A being non empty set for u being UnOp of A for o being BinOp of A holds ( u is_distributive_wrt o iff for a, b being Element of A holds u . (o . (a,b)) = o . ((u . a),(u . b)) );
d21_binop_1:: for X, Y, Z being set for f1, f2 being Function of [:X,Y:],Z holds ( f1 = f2 iff for x, y being set st x in X & y in Y holds f1 . (x,y) = f2 . (x,y) );
d3_binop_1:: for A being set for o being BinOp of A holds ( o is associative iff for a, b, c being Element of A holds o . (a,(o . (b,c))) = o . ((o . (a,b)),c) );
d4_binop_1:: for A being set for o being BinOp of A holds ( o is idempotent iff for a being Element of A holds o . (a,a) = a );
d5_binop_1:: for A being set for e being Element of A for o being BinOp of A holds ( e is_a_left_unity_wrt o iff for a being Element of A holds o . (e,a) = a );
d6_binop_1:: for A being set for e being Element of A for o being BinOp of A holds ( e is_a_right_unity_wrt o iff for a being Element of A holds o . (a,e) = a );
d7_binop_1:: for A being set for e being Element of A for o being BinOp of A holds ( e is_a_unity_wrt o iff ( e is_a_left_unity_wrt o & e is_a_right_unity_wrt o ) );
d8_binop_1:: for A being set for o being BinOp of A st ex e being Element of A st e is_a_unity_wrt o holds for b3 being Element of A holds ( b3 = the_unity_wrt o iff b3 is_a_unity_wrt o );
d9_binop_1:: for A being set for o9, o being BinOp of A holds ( o9 is_left_distributive_wrt o iff for a, b, c being Element of A holds o9 . (a,(o . (b,c))) = o . ((o9 . (a,b)),(o9 . (a,c))) );
s1_binop_1:: scheme FuncEx2{ F1() -> set , F2() -> set , F3() -> set , P1[ set , set , set ] } : ex f being Function of [:F1(),F2():],F3() st for x, y being set st x in F1() & y in F2() holds P1[x,y,f . (x,y)] provided A1: for x, y being set st x in F1() & y in F2() holds ex z being set st ( z in F3() & P1[x,y,z] )
s2_binop_1:: scheme Lambda2{ F1() -> set , F2() -> set , F3() -> set , F4( set , set ) -> set } : ex f being Function of [:F1(),F2():],F3() st for x, y being set st x in F1() & y in F2() holds f . (x,y) = F4(x,y) provided A1: for x, y being set st x in F1() & y in F2() holds F4(x,y) in F3()
s3_binop_1:: scheme FuncEx2D{ F1() -> non empty set , F2() -> non empty set , F3() -> non empty set , P1[ set , set , set ] } : ex f being Function of [:F1(),F2():],F3() st for x being Element of F1() for y being Element of F2() holds P1[x,y,f . (x,y)] provided A1: for x being Element of F1() for y being Element of F2() ex z being Element of F3() st P1[x,y,z]
s4_binop_1:: scheme Lambda2D{ F1() -> non empty set , F2() -> non empty set , F3() -> non empty set , F4( Element of F1(), Element of F2()) -> Element of F3() } : ex f being Function of [:F1(),F2():],F3() st for x being Element of F1() for y being Element of F2() holds f . (x,y) = F4(x,y)
s5_binop_1:: scheme PartFuncEx2{ F1() -> set , F2() -> set , F3() -> set , P1[ set , set , set ] } : ex f being PartFunc of [:F1(),F2():],F3() st ( ( for x, y being set holds ( [x,y] in dom f iff ( x in F1() & y in F2() & ex z being set st P1[x,y,z] ) ) ) & ( for x, y being set st [x,y] in dom f holds P1[x,y,f . (x,y)] ) ) provided A1: for x, y, z being set st x in F1() & y in F2() & P1[x,y,z] holds z in F3() and A2: for x, y, z1, z2 being set st x in F1() & y in F2() & P1[x,y,z1] & P1[x,y,z2] holds z1 = z2
s6_binop_1:: scheme LambdaR2{ F1() -> set , F2() -> set , F3() -> set , F4( set , set ) -> set , P1[ set , set ] } : ex f being PartFunc of [:F1(),F2():],F3() st ( ( for x, y being set holds ( [x,y] in dom f iff ( x in F1() & y in F2() & P1[x,y] ) ) ) & ( for x, y being set st [x,y] in dom f holds f . (x,y) = F4(x,y) ) ) provided A1: for x, y being set st P1[x,y] holds F4(x,y) in F3()
s7_binop_1:: scheme PartLambda2{ F1() -> set , F2() -> set , F3() -> set , F4( set , set ) -> set , P1[ set , set ] } : ex f being PartFunc of [:F1(),F2():],F3() st ( ( for x, y being set holds ( [x,y] in dom f iff ( x in F1() & y in F2() & P1[x,y] ) ) ) & ( for x, y being set st [x,y] in dom f holds f . (x,y) = F4(x,y) ) ) provided A1: for x, y being set st x in F1() & y in F2() & P1[x,y] holds F4(x,y) in F3()
s8_binop_1:: scheme Sch8{ F1() -> non empty set , F2() -> non empty set , F3() -> set , F4( set , set ) -> set , P1[ set , set ] } : ex f being PartFunc of [:F1(),F2():],F3() st ( ( for x being Element of F1() for y being Element of F2() holds ( [x,y] in dom f iff P1[x,y] ) ) & ( for x being Element of F1() for y being Element of F2() st [x,y] in dom f holds f . (x,y) = F4(x,y) ) ) provided A1: for x being Element of F1() for y being Element of F2() st P1[x,y] holds F4(x,y) in F3()
t1_binop_1:: for X, Y, Z being set for f1, f2 being Function of [:X,Y:],Z st ( for x, y being set st x in X & y in Y holds f1 . (x,y) = f2 . (x,y) ) holds f1 = f2
t10_binop_1:: for A being set for o being BinOp of A for e1, e2 being Element of A st e1 is_a_unity_wrt o & e2 is_a_unity_wrt o holds e1 = e2
t11_binop_1:: for A being set for o9, o being BinOp of A holds ( o9 is_distributive_wrt o iff for a, b, c being Element of A holds ( o9 . (a,(o . (b,c))) = o . ((o9 . (a,b)),(o9 . (a,c))) & o9 . ((o . (a,b)),c) = o . ((o9 . (a,c)),(o9 . (b,c))) ) )
t12_binop_1:: for A being non empty set for o, o9 being BinOp of A st o9 is commutative holds ( o9 is_distributive_wrt o iff for a, b, c being Element of A holds o9 . (a,(o . (b,c))) = o . ((o9 . (a,b)),(o9 . (a,c))) )
t13_binop_1:: for A being non empty set for o, o9 being BinOp of A st o9 is commutative holds ( o9 is_distributive_wrt o iff for a, b, c being Element of A holds o9 . ((o . (a,b)),c) = o . ((o9 . (a,c)),(o9 . (b,c))) )
t14_binop_1:: for A being non empty set for o, o9 being BinOp of A st o9 is commutative holds ( o9 is_distributive_wrt o iff o9 is_left_distributive_wrt o )
t15_binop_1:: for A being non empty set for o, o9 being BinOp of A st o9 is commutative holds ( o9 is_distributive_wrt o iff o9 is_right_distributive_wrt o )
t16_binop_1:: for A being non empty set for o, o9 being BinOp of A st o9 is commutative holds ( o9 is_right_distributive_wrt o iff o9 is_left_distributive_wrt o )
t17_binop_1:: for X, Y, Z, x, y being set for f being Function of [:X,Y:],Z st x in X & y in Y & Z <> {} holds f . (x,y) in Z
t18_binop_1:: for x, y, X, Y, Z being set for f being Function of [:X,Y:],Z for g being Function st Z <> {} & x in X & y in Y holds (g * f) . (x,y) = g . (f . (x,y))
t19_binop_1:: for X, Y being set for f being Function st dom f = [:X,Y:] holds ( f is constant iff for x1, x2, y1, y2 being set st x1 in X & x2 in X & y1 in Y & y2 in Y holds f . (x1,y1) = f . (x2,y2) )
t2_binop_1:: for X, Y, Z being set for f1, f2 being Function of [:X,Y:],Z st ( for a being Element of X for b being Element of Y holds f1 . (a,b) = f2 . (a,b) ) holds f1 = f2
t20_binop_1:: for X, Y, Z being set for f1, f2 being PartFunc of [:X,Y:],Z st dom f1 = dom f2 & ( for x, y being set st [x,y] in dom f1 holds f1 . (x,y) = f2 . (x,y) ) holds f1 = f2
t3_binop_1:: for A being set for o being BinOp of A for e being Element of A holds ( e is_a_unity_wrt o iff for a being Element of A holds ( o . (e,a) = a & o . (a,e) = a ) )
t4_binop_1:: for A being set for o being BinOp of A for e being Element of A st o is commutative holds ( e is_a_unity_wrt o iff for a being Element of A holds o . (e,a) = a )
t5_binop_1:: for A being set for o being BinOp of A for e being Element of A st o is commutative holds ( e is_a_unity_wrt o iff for a being Element of A holds o . (a,e) = a )
t6_binop_1:: for A being set for o being BinOp of A for e being Element of A st o is commutative holds ( e is_a_unity_wrt o iff e is_a_left_unity_wrt o )
t7_binop_1:: for A being set for o being BinOp of A for e being Element of A st o is commutative holds ( e is_a_unity_wrt o iff e is_a_right_unity_wrt o )
t8_binop_1:: for A being set for o being BinOp of A for e being Element of A st o is commutative holds ( e is_a_left_unity_wrt o iff e is_a_right_unity_wrt o )
t9_binop_1:: for A being set for o being BinOp of A for e1, e2 being Element of A st e1 is_a_left_unity_wrt o & e2 is_a_right_unity_wrt o holds e1 = e2
d1_binop_2:: for b1 being UnOp of COMPLEX holds ( b1 = compcomplex iff for c being complex number holds b1 . c = - c );
d10_binop_2:: for b1 being BinOp of REAL holds ( b1 = diffreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 - r2 );
d11_binop_2:: for b1 being BinOp of REAL holds ( b1 = multreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 * r2 );
d12_binop_2:: for b1 being BinOp of REAL holds ( b1 = divreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 / r2 );
d13_binop_2:: for b1 being UnOp of RAT holds ( b1 = comprat iff for w being rational number holds b1 . w = - w );
d14_binop_2:: for b1 being UnOp of RAT holds ( b1 = invrat iff for w being rational number holds b1 . w = w " );
d15_binop_2:: for b1 being BinOp of RAT holds ( b1 = addrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 + w2 );
d16_binop_2:: for b1 being BinOp of RAT holds ( b1 = diffrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 - w2 );
d17_binop_2:: for b1 being BinOp of RAT holds ( b1 = multrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 * w2 );
d18_binop_2:: for b1 being BinOp of RAT holds ( b1 = divrat iff for w1, w2 being rational number holds b1 . (w1,w2) = w1 / w2 );
d19_binop_2:: for b1 being UnOp of INT holds ( b1 = compint iff for i being integer number holds b1 . i = - i );
d2_binop_2:: for b1 being UnOp of COMPLEX holds ( b1 = invcomplex iff for c being complex number holds b1 . c = c " );
d20_binop_2:: for b1 being BinOp of INT holds ( b1 = addint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 + i2 );
d21_binop_2:: for b1 being BinOp of INT holds ( b1 = diffint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 - i2 );
d22_binop_2:: for b1 being BinOp of INT holds ( b1 = multint iff for i1, i2 being integer number holds b1 . (i1,i2) = i1 * i2 );
d23_binop_2:: for b1 being BinOp of NAT holds ( b1 = addnat iff for n1, n2 being Nat holds b1 . (n1,n2) = n1 + n2 );
d24_binop_2:: for b1 being BinOp of NAT holds ( b1 = multnat iff for n1, n2 being Nat holds b1 . (n1,n2) = n1 * n2 );
d3_binop_2:: for b1 being BinOp of COMPLEX holds ( b1 = addcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 + c2 );
d4_binop_2:: for b1 being BinOp of COMPLEX holds ( b1 = diffcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 - c2 );
d5_binop_2:: for b1 being BinOp of COMPLEX holds ( b1 = multcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 * c2 );
d6_binop_2:: for b1 being BinOp of COMPLEX holds ( b1 = divcomplex iff for c1, c2 being complex number holds b1 . (c1,c2) = c1 / c2 );
d7_binop_2:: for b1 being UnOp of REAL holds ( b1 = compreal iff for r being real number holds b1 . r = - r );
d8_binop_2:: for b1 being UnOp of REAL holds ( b1 = invreal iff for r being real number holds b1 . r = r " );
d9_binop_2:: for b1 being BinOp of REAL holds ( b1 = addreal iff for r1, r2 being real number holds b1 . (r1,r2) = r1 + r2 );
s1_binop_2:: scheme FuncDefUniq{ F1() -> non empty set , F2() -> non empty set , F3( Element of F1()) -> set } : for f1, f2 being Function of F1(),F2() st ( for x being Element of F1() holds f1 . x = F3(x) ) & ( for x being Element of F1() holds f2 . x = F3(x) ) holds f1 = f2
s10_binop_2:: scheme WBinOpDefuniq{ F1( rational number , rational number ) -> set } : for o1, o2 being BinOp of RAT st ( for a, b being rational number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being rational number holds o2 . (a,b) = F1(a,b) ) holds o1 = o2
s11_binop_2:: scheme IBinOpDefuniq{ F1( integer number , integer number ) -> set } : for o1, o2 being BinOp of INT st ( for a, b being integer number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being integer number holds o2 . (a,b) = F1(a,b) ) holds o1 = o2
s12_binop_2:: scheme NBinOpDefuniq{ F1( Nat, Nat) -> set } : for o1, o2 being BinOp of NAT st ( for a, b being Nat holds o1 . (a,b) = F1(a,b) ) & ( for a, b being Nat holds o2 . (a,b) = F1(a,b) ) holds o1 = o2
s13_binop_2:: scheme CLambda2D{ F1( complex number , complex number ) -> complex number } : ex f being Function of [:COMPLEX,COMPLEX:],COMPLEX st for x, y being complex number holds f . (x,y) = F1(x,y)
s14_binop_2:: scheme RLambda2D{ F1( real number , real number ) -> real number } : ex f being Function of [:REAL,REAL:],REAL st for x, y being real number holds f . (x,y) = F1(x,y)
s15_binop_2:: scheme WLambda2D{ F1( rational number , rational number ) -> rational number } : ex f being Function of [:RAT,RAT:],RAT st for x, y being rational number holds f . (x,y) = F1(x,y)
s16_binop_2:: scheme ILambda2D{ F1( integer number , integer number ) -> integer number } : ex f being Function of [:INT,INT:],INT st for x, y being integer number holds f . (x,y) = F1(x,y)
s17_binop_2:: scheme NLambda2D{ F1( Nat, Nat) -> Nat } : ex f being Function of [:NAT,NAT:],NAT st for x, y being Nat holds f . (x,y) = F1(x,y)
s18_binop_2:: scheme CLambdaD{ F1( complex number ) -> complex number } : ex f being Function of COMPLEX,COMPLEX st for x being complex number holds f . x = F1(x)
s19_binop_2:: scheme RLambdaD{ F1( real number ) -> real number } : ex f being Function of REAL,REAL st for x being real number holds f . x = F1(x)
s2_binop_2:: scheme BinOpDefuniq{ F1() -> non empty set , F2( Element of F1(), Element of F1()) -> set } : for o1, o2 being BinOp of F1() st ( for a, b being Element of F1() holds o1 . (a,b) = F2(a,b) ) & ( for a, b being Element of F1() holds o2 . (a,b) = F2(a,b) ) holds o1 = o2
s20_binop_2:: scheme WLambdaD{ F1( rational number ) -> rational number } : ex f being Function of RAT,RAT st for x being rational number holds f . x = F1(x)
s21_binop_2:: scheme ILambdaD{ F1( integer number ) -> integer number } : ex f being Function of INT,INT st for x being integer number holds f . x = F1(x)
s22_binop_2:: scheme NLambdaD{ F1( Nat) -> Nat } : ex f being Function of NAT,NAT st for x being Nat holds f . x = F1(x)
s3_binop_2:: scheme CFuncDefUniq{ F1( complex number ) -> set } : for f1, f2 being Function of COMPLEX,COMPLEX st ( for x being complex number holds f1 . x = F1(x) ) & ( for x being complex number holds f2 . x = F1(x) ) holds f1 = f2
s4_binop_2:: scheme RFuncDefUniq{ F1( real number ) -> set } : for f1, f2 being Function of REAL,REAL st ( for x being real number holds f1 . x = F1(x) ) & ( for x being real number holds f2 . x = F1(x) ) holds f1 = f2
s5_binop_2:: scheme WFuncDefUniq{ F1( rational number ) -> set } : for f1, f2 being Function of RAT,RAT st ( for x being rational number holds f1 . x = F1(x) ) & ( for x being rational number holds f2 . x = F1(x) ) holds f1 = f2
s6_binop_2:: scheme IFuncDefUniq{ F1( integer number ) -> set } : for f1, f2 being Function of INT,INT st ( for x being integer number holds f1 . x = F1(x) ) & ( for x being integer number holds f2 . x = F1(x) ) holds f1 = f2
s7_binop_2:: scheme NFuncDefUniq{ F1( Nat) -> set } : for f1, f2 being Function of NAT,NAT st ( for x being Nat holds f1 . x = F1(x) ) & ( for x being Nat holds f2 . x = F1(x) ) holds f1 = f2
s8_binop_2:: scheme CBinOpDefuniq{ F1( complex number , complex number ) -> set } : for o1, o2 being BinOp of COMPLEX st ( for a, b being complex number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being complex number holds o2 . (a,b) = F1(a,b) ) holds o1 = o2
s9_binop_2:: scheme RBinOpDefuniq{ F1( real number , real number ) -> set } : for o1, o2 being BinOp of REAL st ( for a, b being real number holds o1 . (a,b) = F1(a,b) ) & ( for a, b being real number holds o2 . (a,b) = F1(a,b) ) holds o1 = o2
t1_binop_2:: the_unity_wrt addcomplex = 0 by Lm2, BINOP_1:def_8;
t10_binop_2:: the_unity_wrt multnat = 1 by Lm12, BINOP_1:def_8;
t2_binop_2:: the_unity_wrt addreal = 0 by Lm3, BINOP_1:def_8;
t3_binop_2:: the_unity_wrt addrat = 0 by Lm4, BINOP_1:def_8;
t4_binop_2:: the_unity_wrt addint = 0 by Lm5, BINOP_1:def_8;
t5_binop_2:: the_unity_wrt addnat = 0 by Lm6, BINOP_1:def_8;
t6_binop_2:: the_unity_wrt multcomplex = 1 by Lm8, BINOP_1:def_8;
t7_binop_2:: the_unity_wrt multreal = 1 by Lm9, BINOP_1:def_8;
t8_binop_2:: the_unity_wrt multrat = 1 by Lm10, BINOP_1:def_8;
t9_binop_2:: the_unity_wrt multint = 1 by Lm11, BINOP_1:def_8;
d1_bintree1:: for D being non empty set for t being DecoratedTree of D holds root-label t = t . {};
d2_bintree1:: for IT being Tree holds ( IT is binary iff for t being Element of IT st not t in Leaves IT holds succ t = {(t ^ <*0*>),(t ^ <*1*>)} );
d3_bintree1:: for IT being DecoratedTree holds ( IT is binary iff dom IT is binary );
d4_bintree1:: for IT being non empty DTConstrStr holds ( IT is binary iff for s being Symbol of IT for p being FinSequence st s ==> p holds ex x1, x2 being Symbol of IT st p = <*x1,x2*> );
s1_bintree1:: scheme BinDTConstrStrEx{ F1() -> non empty set , P1[ set , set , set ] } : ex G being non empty strict binary DTConstrStr st ( the carrier of G = F1() & ( for x, y, z being Symbol of G holds ( x ==> <*y,z*> iff P1[x,y,z] ) ) )
s2_bintree1:: scheme BinDTConstrInd{ F1() -> non empty with_terminals with_nonterminals binary DTConstrStr , P1[ set ] } : for t being Element of TS F1() holds P1[t] provided A1: for s being Terminal of F1() holds P1[ root-tree s] and A2: for nt being NonTerminal of F1() for tl, tr being Element of TS F1() st nt ==> <*(root-label tl),(root-label tr)*> & P1[tl] & P1[tr] holds P1[nt -tree (tl,tr)]
s3_bintree1:: scheme BinDTConstrIndDef{ F1() -> non empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F2() -> non empty set , F3( set ) -> Element of F2(), F4( set , set , set , set , set ) -> Element of F2() } : ex f being Function of (TS F1()),F2() st ( ( for t being Terminal of F1() holds f . (root-tree t) = F3(t) ) & ( for nt being NonTerminal of F1() for tl, tr being Element of TS F1() for rtl, rtr being Symbol of F1() st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds for xl, xr being Element of F2() st xl = f . tl & xr = f . tr holds f . (nt -tree (tl,tr)) = F4(nt,rtl,rtr,xl,xr) ) )
s4_bintree1:: scheme BinDTConstrUniqDef{ F1() -> non empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F2() -> non empty set , F3() -> Function of (TS F1()),F2(), F4() -> Function of (TS F1()),F2(), F5( set ) -> Element of F2(), F6( set , set , set , set , set ) -> Element of F2() } : F3() = F4() provided A1: ( ( for t being Terminal of F1() holds F3() . (root-tree t) = F5(t) ) & ( for nt being NonTerminal of F1() for tl, tr being Element of TS F1() for rtl, rtr being Symbol of F1() st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds for xl, xr being Element of F2() st xl = F3() . tl & xr = F3() . tr holds F3() . (nt -tree (tl,tr)) = F6(nt,rtl,rtr,xl,xr) ) ) and A2: ( ( for t being Terminal of F1() holds F4() . (root-tree t) = F5(t) ) & ( for nt being NonTerminal of F1() for tl, tr being Element of TS F1() for rtl, rtr being Symbol of F1() st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds for xl, xr being Element of F2() st xl = F4() . tl & xr = F4() . tr holds F4() . (nt -tree (tl,tr)) = F6(nt,rtl,rtr,xl,xr) ) )
s5_bintree1:: scheme BinDTCDefLambda{ F1() -> non empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F2() -> non empty set , F3() -> non empty set , F4( set , set ) -> Element of F3(), F5( set , set , set , set ) -> Element of F3() } : ex f being Function of (TS F1()),(Funcs (F2(),F3())) st ( ( for t being Terminal of F1() ex g being Function of F2(),F3() st ( g = f . (root-tree t) & ( for a being Element of F2() holds g . a = F4(t,a) ) ) ) & ( for nt being NonTerminal of F1() for t1, t2 being Element of TS F1() for rtl, rtr being Symbol of F1() st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds ex g, f1, f2 being Function of F2(),F3() st ( g = f . (nt -tree (t1,t2)) & f1 = f . t1 & f2 = f . t2 & ( for a being Element of F2() holds g . a = F5(nt,f1,f2,a) ) ) ) )
s6_bintree1:: scheme BinDTCDefLambdaUniq{ F1() -> non empty with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr , F2() -> non empty set , F3() -> non empty set , F4() -> Function of (TS F1()),(Funcs (F2(),F3())), F5() -> Function of (TS F1()),(Funcs (F2(),F3())), F6( set , set ) -> Element of F3(), F7( set , set , set , set ) -> Element of F3() } : F4() = F5() provided A1: ( ( for t being Terminal of F1() ex g being Function of F2(),F3() st ( g = F4() . (root-tree t) & ( for a being Element of F2() holds g . a = F6(t,a) ) ) ) & ( for nt being NonTerminal of F1() for t1, t2 being Element of TS F1() for rtl, rtr being Symbol of F1() st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds ex g, f1, f2 being Function of F2(),F3() st ( g = F4() . (nt -tree (t1,t2)) & f1 = F4() . t1 & f2 = F4() . t2 & ( for a being Element of F2() holds g . a = F7(nt,f1,f2,a) ) ) ) ) and A2: ( ( for t being Terminal of F1() ex g being Function of F2(),F3() st ( g = F5() . (root-tree t) & ( for a being Element of F2() holds g . a = F6(t,a) ) ) ) & ( for nt being NonTerminal of F1() for t1, t2 being Element of TS F1() for rtl, rtr being Symbol of F1() st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds ex g, f1, f2 being Function of F2(),F3() st ( g = F5() . (nt -tree (t1,t2)) & f1 = F5() . t1 & f2 = F5() . t2 & ( for a being Element of F2() holds g . a = F7(nt,f1,f2,a) ) ) ) )
t1_bintree1:: for D being non empty set for t being DecoratedTree of D holds roots <*t*> = <*(root-label t)*> by DTCONSTR:4;
t10_bintree1:: for G being non empty with_terminals with_nonterminals binary DTConstrStr for ts being FinSequence of TS G for nt being Symbol of G st nt ==> roots ts holds ( nt is NonTerminal of G & dom ts = {1,2} & 1 in dom ts & 2 in dom ts & ex tl, tr being Element of TS G st ( roots ts = <*(root-label tl),(root-label tr)*> & tl = ts . 1 & tr = ts . 2 & nt -tree ts = nt -tree (tl,tr) & tl in rng ts & tr in rng ts ) )
t2_bintree1:: for D being non empty set for t1, t2 being DecoratedTree of D holds roots <*t1,t2*> = <*(root-label t1),(root-label t2)*> by DTCONSTR:6;
t3_bintree1:: for T being Tree for t being Element of T holds ( succ t = {} iff t in Leaves T )
t4_bintree1:: elementary_tree 0 is binary ;
t5_bintree1:: elementary_tree 2 is binary ;
t6_bintree1:: for T0, T1 being Tree for t being Element of tree (T0,T1) holds ( ( for p being Element of T0 st t = <*0*> ^ p holds ( t in Leaves (tree (T0,T1)) iff p in Leaves T0 ) ) & ( for p being Element of T1 st t = <*1*> ^ p holds ( t in Leaves (tree (T0,T1)) iff p in Leaves T1 ) ) )
t7_bintree1:: for T0, T1 being Tree for t being Element of tree (T0,T1) holds ( ( t = {} implies succ t = {(t ^ <*0*>),(t ^ <*1*>)} ) & ( for p being Element of T0 st t = <*0*> ^ p holds for sp being FinSequence holds ( sp in succ p iff <*0*> ^ sp in succ t ) ) & ( for p being Element of T1 st t = <*1*> ^ p holds for sp being FinSequence holds ( sp in succ p iff <*1*> ^ sp in succ t ) ) )
t8_bintree1:: for T1, T2 being Tree holds ( ( T1 is binary & T2 is binary ) iff tree (T1,T2) is binary )
t9_bintree1:: for T1, T2 being DecoratedTree for x being set holds ( ( T1 is binary & T2 is binary ) iff x -tree (T1,T2) is binary )
d1_bintree2:: for T being binary Tree for n being non empty Nat for b3 being Function of (T -level n),NAT holds ( b3 = NumberOnLevel (n,T) iff for t being Element of T st t in T -level n holds for F being Element of n -tuples_on BOOLEAN st F = Rev t holds b3 . t = (Absval F) + 1 );
d2_bintree2:: for T being Tree holds ( T is full iff T = {0,1} * );
d3_bintree2:: for T being full Tree for n being non empty Nat holds FinSeqLevel (n,T) = (NumberOnLevel (n,T)) " ;
s1_bintree2:: scheme DecoratedBinTreeEx{ F1() -> non empty set , F2() -> Element of F1(), P1[ set , set , set ] } : ex D being binary DecoratedTree of F1() st ( dom D = {0,1} * & D . {} = F2() & ( for x being Node of D holds P1[D . x,D . (x ^ <*0*>),D . (x ^ <*1*>)] ) ) provided A1: for a being Element of F1() ex b, c being Element of F1() st P1[a,b,c]
s2_bintree2:: scheme DecoratedBinTreeEx1{ F1() -> non empty set , F2() -> Element of F1(), P1[ set , set ], P2[ set , set ] } : ex D being binary DecoratedTree of F1() st ( dom D = {0,1} * & D . {} = F2() & ( for x being Node of D holds ( P1[D . x,D . (x ^ <*0*>)] & P2[D . x,D . (x ^ <*1*>)] ) ) ) provided A1: for a being Element of F1() ex b being Element of F1() st P1[a,b] and A2: for a being Element of F1() ex b being Element of F1() st P2[a,b]
t1_bintree2:: for D being set for p being FinSequence for n being Element of NAT st p in D * holds p | (Seg n) in D *
t10_bintree2:: for T being Tree st T = {0,1} * holds for n being Nat holds 0* n in T -level n
t11_bintree2:: for T being Tree st T = {0,1} * holds for n being non empty Element of NAT for y being Element of n -tuples_on BOOLEAN holds y in T -level n
t12_bintree2:: for T being full Tree for n being non empty Nat holds Seg (2 to_power n) c= rng (NumberOnLevel (n,T))
t13_bintree2:: for T being full Tree for n being non empty Element of NAT holds (NumberOnLevel (n,T)) . (0* n) = 1
t14_bintree2:: for T being full Tree for n being non empty Element of NAT for y being Element of n -tuples_on BOOLEAN st y = 0* n holds (NumberOnLevel (n,T)) . ('not' y) = 2 to_power n
t15_bintree2:: for T being full Tree for n being non empty Element of NAT holds (FinSeqLevel (n,T)) . 1 = 0* n
t16_bintree2:: for T being full Tree for n being non empty Element of NAT for y being Element of n -tuples_on BOOLEAN st y = 0* n holds (FinSeqLevel (n,T)) . (2 to_power n) = 'not' y
t17_bintree2:: for T being full Tree for n being non empty Element of NAT for i being Element of NAT st i in Seg (2 to_power n) holds (FinSeqLevel (n,T)) . i = Rev (n -BinarySequence (i -' 1))
t18_bintree2:: for T being full Tree for n being Element of NAT holds card (T -level n) = 2 to_power n
t19_bintree2:: for T being full Tree for n being non empty Element of NAT holds len (FinSeqLevel (n,T)) = 2 to_power n
t2_bintree2:: for T being binary Tree for t being Element of T holds t is FinSequence of BOOLEAN
t20_bintree2:: for T being full Tree for n being non empty Element of NAT holds dom (FinSeqLevel (n,T)) = Seg (2 to_power n)
t21_bintree2:: for T being full Tree for n being non empty Element of NAT holds rng (FinSeqLevel (n,T)) = T -level n
t22_bintree2:: for T being full Tree holds (FinSeqLevel (1,T)) . 1 = <*0*>
t23_bintree2:: for T being full Tree holds (FinSeqLevel (1,T)) . 2 = <*1*>
t24_bintree2:: for T being full Tree for n, i being non empty Element of NAT st i <= 2 to_power (n + 1) holds for F being Element of n -tuples_on BOOLEAN st F = (FinSeqLevel (n,T)) . ((i + 1) div 2) holds (FinSeqLevel ((n + 1),T)) . i = F ^ <*((i + 1) mod 2)*>
t3_bintree2:: for T being Tree st T = {0,1} * holds T is binary
t4_bintree2:: for T being Tree st T = {0,1} * holds Leaves T = {}
t5_bintree2:: for T being binary Tree for n being Element of NAT for t being Element of T st t in T -level n holds t is Element of n -tuples_on BOOLEAN
t6_bintree2:: for T being Tree st ( for t being Element of T holds succ t = {(t ^ <*0*>),(t ^ <*1*>)} ) holds Leaves T = {}
t7_bintree2:: for T being Tree st ( for t being Element of T holds succ t = {(t ^ <*0*>),(t ^ <*1*>)} ) holds T is binary
t8_bintree2:: for T being Tree holds ( T = {0,1} * iff for t being Element of T holds succ t = {(t ^ <*0*>),(t ^ <*1*>)} )
t9_bintree2:: {0,1} * is Tree
d1_birkhoff:: for S being non empty non void ManySortedSign for X being V2() ManySortedSet of the carrier of S for A being non-empty MSAlgebra over S for F being ManySortedFunction of X, the Sorts of A for b5 being ManySortedFunction of (FreeMSA X),A holds ( b5 = F -hash iff ( b5 is_homomorphism FreeMSA X,A & b5 || (FreeGen X) = F ** (Reverse X) ) );
s1_birkhoff:: scheme ExFreeAlg1{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), P1[ set ] } : ex A being strict non-empty MSAlgebra over F1() ex F being ManySortedFunction of F2(),A st ( P1[A] & F is_epimorphism F2(),A & ( for B being non-empty MSAlgebra over F1() for G being ManySortedFunction of F2(),B st G is_homomorphism F2(),B & P1[B] holds ex H being ManySortedFunction of A,B st ( H is_homomorphism A,B & H ** F = G & ( for K being ManySortedFunction of A,B st K ** F = G holds H = K ) ) ) ) provided A1: for A, B being non-empty MSAlgebra over F1() st A,B are_isomorphic & P1[A] holds P1[B] and A2: for A being non-empty MSAlgebra over F1() for B being strict non-empty MSSubAlgebra of A st P1[A] holds P1[B] and A3: for I being set for F being MSAlgebra-Family of I,F1() st ( for i being set st i in I holds ex A being MSAlgebra over F1() st ( A = F . i & P1[A] ) ) holds P1[ product F]
s10_birkhoff:: scheme FreeInModIsInVar1{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), P1[ set ], P2[ set ] } : P2[F2()] provided A1: for A being non-empty MSAlgebra over F1() holds ( P2[A] iff for s being SortSymbol of F1() for e being Element of (Equations F1()) . s st ( for B being non-empty MSAlgebra over F1() st P1[B] holds B |= e ) holds A |= e ) and A2: P1[F2()]
s11_birkhoff:: scheme FreeInModIsInVar{ F1() -> non empty non void ManySortedSign , F2() -> strict non-empty MSAlgebra over F1(), F3() -> ManySortedFunction of the carrier of F1() --> NAT, the Sorts of F2(), P1[ set ], P2[ set ] } : P1[F2()] provided A1: for A being non-empty MSAlgebra over F1() holds ( P2[A] iff for s being SortSymbol of F1() for e being Element of (Equations F1()) . s st ( for B being non-empty MSAlgebra over F1() st P1[B] holds B |= e ) holds A |= e ) and A2: for C being non-empty MSAlgebra over F1() for G being ManySortedFunction of the carrier of F1() --> NAT, the Sorts of C st P2[C] holds ex H being ManySortedFunction of F2(),C st ( H is_homomorphism F2(),C & H ** F3() = G & ( for K being ManySortedFunction of F2(),C st K is_homomorphism F2(),C & K ** F3() = G holds H = K ) ) and A3: P2[F2()] and A4: for A, B being non-empty MSAlgebra over F1() st A,B are_isomorphic & P1[A] holds P1[B] and A5: for A being non-empty MSAlgebra over F1() for B being strict non-empty MSSubAlgebra of A st P1[A] holds P1[B] and A6: for I being set for F being MSAlgebra-Family of I,F1() st ( for i being set st i in I holds ex A being MSAlgebra over F1() st ( A = F . i & P1[A] ) ) holds P1[ product F]
s12_birkhoff:: scheme Birkhoff{ F1() -> non empty non void ManySortedSign , P1[ set ] } : ex E being EqualSet of F1() st for A being non-empty MSAlgebra over F1() holds ( P1[A] iff A |= E ) provided A1: for A, B being non-empty MSAlgebra over F1() st A,B are_isomorphic & P1[A] holds P1[B] and A2: for A being non-empty MSAlgebra over F1() for B being strict non-empty MSSubAlgebra of A st P1[A] holds P1[B] and A3: for A being non-empty MSAlgebra over F1() for R being MSCongruence of A st P1[A] holds P1[ QuotMSAlg (A,R)] and A4: for I being set for F being MSAlgebra-Family of I,F1() st ( for i being set st i in I holds ex A being MSAlgebra over F1() st ( A = F . i & P1[A] ) ) holds P1[ product F]
s2_birkhoff:: scheme ExFreeAlg2{ F1() -> non empty non void ManySortedSign , F2() -> V2() ManySortedSet of the carrier of F1(), P1[ set ] } : ex A being strict non-empty MSAlgebra over F1() ex F being ManySortedFunction of F2(), the Sorts of A st ( P1[A] & ( for B being non-empty MSAlgebra over F1() for G being ManySortedFunction of F2(), the Sorts of B st P1[B] holds ex H being ManySortedFunction of A,B st ( H is_homomorphism A,B & H ** F = G & ( for K being ManySortedFunction of A,B st K is_homomorphism A,B & K ** F = G holds H = K ) ) ) ) provided A1: for A, B being non-empty MSAlgebra over F1() st A,B are_isomorphic & P1[A] holds P1[B] and A2: for A being non-empty MSAlgebra over F1() for B being strict non-empty MSSubAlgebra of A st P1[A] holds P1[B] and A3: for I being set for F being MSAlgebra-Family of I,F1() st ( for i being set st i in I holds ex A being MSAlgebra over F1() st ( A = F . i & P1[A] ) ) holds P1[ product F]
s3_birkhoff:: scheme Exhash{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), F3() -> non-empty MSAlgebra over F1(), F4() -> ManySortedFunction of the carrier of F1() --> NAT, the Sorts of F2(), F5() -> ManySortedFunction of the carrier of F1() --> NAT, the Sorts of F3(), P1[ set ] } : ex H being ManySortedFunction of F2(),F3() st ( H is_homomorphism F2(),F3() & F5() -hash = H ** (F4() -hash) ) provided A1: P1[F3()] and A2: for C being non-empty MSAlgebra over F1() for G being ManySortedFunction of the carrier of F1() --> NAT, the Sorts of C st P1[C] holds ex h being ManySortedFunction of F2(),C st ( h is_homomorphism F2(),C & G = h ** F4() )
s4_birkhoff:: scheme EqTerms{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), F3() -> ManySortedFunction of the carrier of F1() --> NAT, the Sorts of F2(), F4() -> SortSymbol of F1(), F5() -> Element of the Sorts of (TermAlg F1()) . F4(), F6() -> Element of the Sorts of (TermAlg F1()) . F4(), P1[ set ] } : for B being non-empty MSAlgebra over F1() st P1[B] holds B |= F5() '=' F6() provided A1: ((F3() -hash) . F4()) . F5() = ((F3() -hash) . F4()) . F6() and A2: for C being non-empty MSAlgebra over F1() for G being ManySortedFunction of the carrier of F1() --> NAT, the Sorts of C st P1[C] holds ex h being ManySortedFunction of F2(),C st ( h is_homomorphism F2(),C & G = h ** F3() )
s5_birkhoff:: scheme FreeIsGen{ F1() -> non empty non void ManySortedSign , F2() -> V2() ManySortedSet of the carrier of F1(), F3() -> strict non-empty MSAlgebra over F1(), F4() -> ManySortedFunction of F2(), the Sorts of F3(), P1[ set ] } : F4() .:.: F2() is V2() GeneratorSet of F3() provided A1: for C being non-empty MSAlgebra over F1() for G being ManySortedFunction of F2(), the Sorts of C st P1[C] holds ex H being ManySortedFunction of F3(),C st ( H is_homomorphism F3(),C & H ** F4() = G & ( for K being ManySortedFunction of F3(),C st K is_homomorphism F3(),C & K ** F4() = G holds H = K ) ) and A2: P1[F3()] and A3: for A being non-empty MSAlgebra over F1() for B being strict non-empty MSSubAlgebra of A st P1[A] holds P1[B]
s6_birkhoff:: scheme Hashisonto{ F1() -> non empty non void ManySortedSign , F2() -> strict non-empty MSAlgebra over F1(), F3() -> ManySortedFunction of the carrier of F1() --> NAT, the Sorts of F2(), P1[ set ] } : F3() -hash is_epimorphism FreeMSA ( the carrier of F1() --> NAT),F2() provided A1: for C being non-empty MSAlgebra over F1() for G being ManySortedFunction of the carrier of F1() --> NAT, the Sorts of C st P1[C] holds ex H being ManySortedFunction of F2(),C st ( H is_homomorphism F2(),C & H ** F3() = G & ( for K being ManySortedFunction of F2(),C st K is_homomorphism F2(),C & K ** F3() = G holds H = K ) ) and A2: P1[F2()] and A3: for A being non-empty MSAlgebra over F1() for B being strict non-empty MSSubAlgebra of A st P1[A] holds P1[B]
s7_birkhoff:: scheme FinGenAlgInVar{ F1() -> non empty non void ManySortedSign , F2() -> strict non-empty finitely-generated MSAlgebra over F1(), F3() -> non-empty MSAlgebra over F1(), F4() -> ManySortedFunction of the carrier of F1() --> NAT, the Sorts of F3(), P1[ set ], P2[ set ] } : P1[F2()] provided A1: P2[F2()] and A2: P1[F3()] and A3: for C being non-empty MSAlgebra over F1() for G being ManySortedFunction of the carrier of F1() --> NAT, the Sorts of C st P2[C] holds ex h being ManySortedFunction of F3(),C st ( h is_homomorphism F3(),C & G = h ** F4() ) and A4: for A, B being non-empty MSAlgebra over F1() st A,B are_isomorphic & P1[A] holds P1[B] and A5: for A being non-empty MSAlgebra over F1() for R being MSCongruence of A st P1[A] holds P1[ QuotMSAlg (A,R)]
s8_birkhoff:: scheme QuotEpi{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), F3() -> non-empty MSAlgebra over F1(), P1[ set ] } : P1[F3()] provided A1: ex H being ManySortedFunction of F2(),F3() st H is_epimorphism F2(),F3() and A2: P1[F2()] and A3: for A, B being non-empty MSAlgebra over F1() st A,B are_isomorphic & P1[A] holds P1[B] and A4: for A being non-empty MSAlgebra over F1() for R being MSCongruence of A st P1[A] holds P1[ QuotMSAlg (A,R)]
s9_birkhoff:: scheme AllFinGen{ F1() -> non empty non void ManySortedSign , F2() -> non-empty MSAlgebra over F1(), P1[ set ] } : P1[F2()] provided A1: for B being strict non-empty finitely-generated MSSubAlgebra of F2() holds P1[B] and A2: for A, B being non-empty MSAlgebra over F1() st A,B are_isomorphic & P1[A] holds P1[B] and A3: for A being non-empty MSAlgebra over F1() for B being strict non-empty MSSubAlgebra of A st P1[A] holds P1[B] and A4: for A being non-empty MSAlgebra over F1() for R being MSCongruence of A st P1[A] holds P1[ QuotMSAlg (A,R)] and A5: for I being set for F being MSAlgebra-Family of I,F1() st ( for i being set st i in I holds ex A being MSAlgebra over F1() st ( A = F . i & P1[A] ) ) holds P1[ product F]
t1_birkhoff:: for S being non empty non void ManySortedSign for A being non-empty MSAlgebra over S for X being V2() ManySortedSet of the carrier of S for F being ManySortedFunction of X, the Sorts of A holds rngs F c= rngs (F -hash)
t1_boole:: for X being set holds X \/ {} = X
t2_boole:: for X being set holds X /\ {} = {}
t3_boole:: for X being set holds X \ {} = X
t4_boole:: for X being set holds {} \ X = {}
t5_boole:: for X being set holds X \+\ {} = X
t6_boole:: for X being set st X is empty holds X = {} by Lm1;
t7_boole:: for x, X being set st x in X holds not X is empty by XBOOLE_0:def_1;
t8_boole:: for X, Y being set st X is empty & X <> Y holds not Y is empty
d1_boolealg:: for L being Lattice for X, Y being Element of L holds X \ Y = X "/\" (Y `);
d2_boolealg:: for L being Lattice for X, Y being Element of L holds X \+\ Y = (X \ Y) "\/" (Y \ X);
d3_boolealg:: for L being Lattice for X, Y being Element of L holds ( X = Y iff ( X [= Y & Y [= X ) );
d4_boolealg:: for L being Lattice for X, Y being Element of L holds ( X meets Y iff X "/\" Y <> Bottom L );
t1_boolealg:: for L being Lattice for X, Y, Z being Element of L st X "\/" Y [= Z holds X [= Z
t10_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st X [= Y & X [= Z & Y "/\" Z = Bottom L holds X = Bottom L by Th9, FILTER_0:7;
t11_boolealg:: for L being 0_Lattice for X, Y being Element of L holds ( X "\/" Y = Bottom L iff ( X = Bottom L & Y = Bottom L ) )
t12_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st X [= Y & Y "/\" Z = Bottom L holds X "/\" Z = Bottom L by LATTICES:9, Th9;
t13_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st X meets Y & Y [= Z holds X meets Z
t14_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st X meets Y "/\" Z holds ( X meets Y & X meets Z )
t15_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st X meets Y \ Z holds X meets Y
t16_boolealg:: for L being 0_Lattice for X being Element of L holds X misses Bottom L
t17_boolealg:: for L being 0_Lattice for X, Z, Y being Element of L st X misses Z & Y [= Z holds X misses Y by Th13;
t18_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st ( X misses Y or X misses Z ) holds X misses Y "/\" Z by Th14;
t19_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st X [= Y & X [= Z & Y misses Z holds X = Bottom L
t2_boolealg:: for L being Lattice for X, Y, Z being Element of L holds X "/\" Y [= X "\/" Z
t20_boolealg:: for L being 0_Lattice for X, Y, Z being Element of L st X misses Y holds Z "/\" X misses Z "/\" Y
t21_boolealg:: for L being B_Lattice for X, Y, Z being Element of L st X \ Y [= Z holds X [= Y "\/" Z
t22_boolealg:: for L being B_Lattice for X, Y, Z being Element of L st X [= Y holds Z \ Y [= Z \ X
t23_boolealg:: for L being B_Lattice for X, Y, Z, V being Element of L st X [= Y & Z [= V holds X \ V [= Y \ Z
t24_boolealg:: for L being B_Lattice for X, Y, Z being Element of L st X [= Y "\/" Z holds X \ Y [= Z
t25_boolealg:: for L being B_Lattice for X, Y being Element of L holds X ` [= (X "/\" Y) `
t26_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X "\/" Y) ` [= X `
t27_boolealg:: for L being B_Lattice for X, Y being Element of L st X [= Y \ X holds X = Bottom L
t28_boolealg:: for L being B_Lattice for X, Y being Element of L st X [= Y holds Y = X "\/" (Y \ X)
t29_boolealg:: for L being B_Lattice for X, Y being Element of L holds ( X \ Y = Bottom L iff X [= Y )
t3_boolealg:: for L being Lattice for X, Z, Y being Element of L st X [= Z holds X \ Y [= Z
t30_boolealg:: for L being B_Lattice for X, Y, Z being Element of L st X [= Y "\/" Z & X "/\" Z = Bottom L holds X [= Y
t31_boolealg:: for L being B_Lattice for X, Y being Element of L holds X "\/" Y = (X \ Y) "\/" Y
t32_boolealg:: for L being B_Lattice for X, Y being Element of L holds X \ (X "\/" Y) = Bottom L by Th29, LATTICES:5;
t33_boolealg:: for L being B_Lattice for X, Y being Element of L holds X \ (X "/\" Y) = X \ Y
t34_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X \ Y) "/\" Y = Bottom L
t35_boolealg:: for L being B_Lattice for X, Y being Element of L holds X "\/" (Y \ X) = X "\/" Y
t36_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X "/\" Y) "\/" (X \ Y) = X
t37_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds X \ (Y \ Z) = (X \ Y) "\/" (X "/\" Z)
t38_boolealg:: for L being B_Lattice for X, Y being Element of L holds X \ (X \ Y) = X "/\" Y
t39_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X "\/" Y) \ Y = X \ Y
t4_boolealg:: for L being Lattice for X, Y, Z being Element of L st X \ Y [= Z & Y \ X [= Z holds X \+\ Y [= Z by FILTER_0:6;
t40_boolealg:: for L being B_Lattice for X, Y being Element of L holds ( X "/\" Y = Bottom L iff X \ Y = X )
t41_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds X \ (Y "\/" Z) = (X \ Y) "/\" (X \ Z)
t42_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds X \ (Y "/\" Z) = (X \ Y) "\/" (X \ Z)
t43_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds X "/\" (Y \ Z) = (X "/\" Y) \ (X "/\" Z)
t44_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X "\/" Y) \ (X "/\" Y) = (X \ Y) "\/" (Y \ X)
t45_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds (X \ Y) \ Z = X \ (Y "\/" Z)
t46_boolealg:: for L being B_Lattice for X, Y being Element of L st X \ Y = Y \ X holds X = Y
t47_boolealg:: for L being B_Lattice for X being Element of L holds X \ (Bottom L) = X
t48_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X \ Y) ` = (X `) "\/" Y
t49_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds ( X meets Y "\/" Z iff ( X meets Y or X meets Z ) )
t5_boolealg:: for L being Lattice for X, Y, Z being Element of L holds ( X = Y "\/" Z iff ( Y [= X & Z [= X & ( for V being Element of L st Y [= V & Z [= V holds X [= V ) ) )
t50_boolealg:: for L being B_Lattice for X, Y being Element of L holds X "/\" Y misses X \ Y
t51_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds ( X misses Y "\/" Z iff ( X misses Y & X misses Z ) ) by Th49;
t52_boolealg:: for L being B_Lattice for X, Y being Element of L holds X \ Y misses Y
t53_boolealg:: for L being B_Lattice for X, Y being Element of L st X misses Y holds (X "\/" Y) \ Y = X
t54_boolealg:: for L being B_Lattice for X, Y being Element of L st (X `) "\/" (Y `) = X "\/" Y & X misses X ` & Y misses Y ` holds ( X = Y ` & Y = X ` )
t55_boolealg:: for L being B_Lattice for X, Y being Element of L st (X `) "\/" (Y `) = X "\/" Y & Y misses X ` & X misses Y ` holds ( X = X ` & Y = Y ` )
t56_boolealg:: for L being B_Lattice for X being Element of L holds X \+\ (Bottom L) = X
t57_boolealg:: for L being B_Lattice for X being Element of L holds X \+\ X = Bottom L by LATTICES:20;
t58_boolealg:: for L being B_Lattice for X, Y being Element of L holds X "/\" Y misses X \+\ Y
t59_boolealg:: for L being B_Lattice for X, Y being Element of L holds X "\/" Y = X \+\ (Y \ X)
t6_boolealg:: for L being Lattice for X, Y, Z being Element of L holds ( X = Y "/\" Z iff ( X [= Y & X [= Z & ( for V being Element of L st V [= Y & V [= Z holds V [= X ) ) )
t60_boolealg:: for L being B_Lattice for X, Y being Element of L holds X \+\ (X "/\" Y) = X \ Y
t61_boolealg:: for L being B_Lattice for X, Y being Element of L holds X "\/" Y = (X \+\ Y) "\/" (X "/\" Y)
t62_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X \+\ Y) \+\ (X "/\" Y) = X "\/" Y
t63_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X \+\ Y) \+\ (X "\/" Y) = X "/\" Y
t64_boolealg:: for L being B_Lattice for X, Y being Element of L holds X \+\ Y = (X "\/" Y) \ (X "/\" Y)
t65_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds (X \+\ Y) \ Z = (X \ (Y "\/" Z)) "\/" (Y \ (X "\/" Z))
t66_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds X \ (Y \+\ Z) = (X \ (Y "\/" Z)) "\/" ((X "/\" Y) "/\" Z)
t67_boolealg:: for L being B_Lattice for X, Y, Z being Element of L holds (X \+\ Y) \+\ Z = X \+\ (Y \+\ Z)
t68_boolealg:: for L being B_Lattice for X, Y being Element of L holds (X \+\ Y) ` = (X "/\" Y) "\/" ((X `) "/\" (Y `))
t7_boolealg:: for L being Lattice for X being Element of L holds ( X meets X iff X <> Bottom L )
t8_boolealg:: for L being D_Lattice for X, Y, Z being Element of L st (X "/\" Y) "\/" (X "/\" Z) = X holds X [= Y "\/" Z
t9_boolealg:: for L being 0_Lattice for X being Element of L st X [= Bottom L holds X = Bottom L
d1_boolmark:: for PTN being PT_net_Str holds Bool_marks_of PTN = Funcs ( the carrier of PTN,BOOLEAN);
d2_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for t being transition of PTN holds ( t is_firable_on M0 iff M0 .: (*' {t}) c= {TRUE} );
d3_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for t being transition of PTN holds Firing (t,M0) = (M0 +* ((*' {t}) --> FALSE)) +* (({t} *') --> TRUE);
d4_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for Q being FinSequence of the carrier' of PTN holds ( Q is_firable_on M0 iff ( Q = {} or ex M being FinSequence of Bool_marks_of PTN st ( len Q = len M & Q /. 1 is_firable_on M0 & M /. 1 = Firing ((Q /. 1),M0) & ( for i being Element of NAT st i < len Q & i > 0 holds ( Q /. (i + 1) is_firable_on M /. i & M /. (i + 1) = Firing ((Q /. (i + 1)),(M /. i)) ) ) ) ) );
d5_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for Q being FinSequence of the carrier' of PTN for b4 being Boolean_marking of PTN holds ( ( Q = {} implies ( b4 = Firing (Q,M0) iff b4 = M0 ) ) & ( not Q = {} implies ( b4 = Firing (Q,M0) iff ex M being FinSequence of Bool_marks_of PTN st ( len Q = len M & b4 = M /. (len M) & M /. 1 = Firing ((Q /. 1),M0) & ( for i being Element of NAT st i < len Q & i > 0 holds M /. (i + 1) = Firing ((Q /. (i + 1)),(M /. i)) ) ) ) ) );
t1_boolmark:: for A, B being non empty set for f being Function of A,B for C being Subset of A for v being Element of B holds f +* (C --> v) is Function of A,B
t10_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for t being transition of PTN holds ( t is_firable_on M0 iff <*t*> is_firable_on M0 )
t11_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for t being transition of PTN holds Firing (t,M0) = Firing (<*t*>,M0)
t12_boolmark:: for PTN being Petri_net for Sd being non empty Subset of the carrier of PTN holds ( Sd is Deadlock-like iff for M0 being Boolean_marking of PTN st M0 .: Sd = {FALSE} holds for Q being FinSequence of the carrier' of PTN st Q is_firable_on M0 holds (Firing (Q,M0)) .: Sd = {FALSE} )
t2_boolmark:: for X, Y being non empty set for A, B being Subset of X for f being Function of X,Y st f .: A misses f .: B holds A misses B
t3_boolmark:: for A, B being set for f being Function for x being set st A misses B holds (f +* (A --> x)) .: B = f .: B
t4_boolmark:: for A being non empty set for y being set for f being Function holds (f +* (A --> y)) .: A = {y}
t5_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for t being transition of PTN for s being place of PTN st s in {t} *' holds (Firing (t,M0)) . s = TRUE
t6_boolmark:: for PTN being Petri_net for Sd being non empty Subset of the carrier of PTN holds ( Sd is Deadlock-like iff for M0 being Boolean_marking of PTN st M0 .: Sd = {FALSE} holds for t being transition of PTN st t is_firable_on M0 holds (Firing (t,M0)) .: Sd = {FALSE} )
t7_boolmark:: for D being non empty set for Q0, Q1 being FinSequence of D for i being Element of NAT st 1 <= i & i <= len Q0 holds (Q0 ^ Q1) /. i = Q0 /. i
t8_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for Q0, Q1 being FinSequence of the carrier' of PTN holds Firing ((Q0 ^ Q1),M0) = Firing (Q1,(Firing (Q0,M0)))
t9_boolmark:: for PTN being Petri_net for M0 being Boolean_marking of PTN for Q0, Q1 being FinSequence of the carrier' of PTN st Q0 ^ Q1 is_firable_on M0 holds ( Q1 is_firable_on Firing (Q0,M0) & Q0 is_firable_on M0 )
d1_bor_cant:: for s, b2 being Real_Sequence holds ( b2 = JSum s iff for d being Nat holds b2 . d = Sum ((- (s . d)) rExpSeq) );
d10_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma holds @lim_inf A = Union (Intersect_Shift_Seq A);
d11_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for b5 being Real_Sequence holds ( b5 = Sum_Shift_Seq (Prob,A) iff for n being Element of NAT holds b5 . n = Sum (Prob * (A ^\ n)) );
d2_bor_cant:: for n1, n2 being Element of NAT for b3 being sequence of NAT holds ( b3 = Special_Function (n1,n2) iff for n being Element of NAT holds b3 . n = IFGT (n,n1,(n + n2),n) );
d3_bor_cant:: for k being Element of NAT for b2 being sequence of NAT holds ( b2 = Special_Function2 k iff for n being Element of NAT holds b2 . n = n + k );
d4_bor_cant:: for k being Element of NAT for b2 being sequence of NAT holds ( b2 = Special_Function3 k iff for n being Element of NAT holds b2 . n = IFGT (n,k,0,1) );
d5_bor_cant:: for n1, n2 being Element of NAT for b3 being sequence of NAT holds ( b3 = Special_Function4 (n1,n2) iff for n being Element of NAT holds b3 . n = IFGT (n,(n1 + 1),(n + n2),n) );
d6_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( A is_all_independent_wrt Prob iff for B being SetSequence of Sigma st ex e being sequence of NAT st ( e is one-to-one & ( for n being Element of NAT holds A . (e . n) = B . n ) ) holds for n being Element of NAT holds (Partial_Product (Prob * B)) . n = Prob . ((Partial_Intersection B) . n) );
d7_bor_cant:: for X being set for A, b3 being SetSequence of X holds ( b3 = Union_Shift_Seq A iff for n being Element of NAT holds b3 . n = Union (A ^\ n) );
d8_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma holds @lim_sup A = @Intersection (Union_Shift_Seq A);
d9_bor_cant:: for X being set for A, b3 being SetSequence of X holds ( b3 = Intersect_Shift_Seq A iff for n being Element of NAT holds b3 . n = Intersection (A ^\ n) );
t1_bor_cant:: for k being Element of NAT for x being Element of REAL st k is odd & x > 0 & x <= 1 holds (((- x) rExpSeq) . (k + 1)) + (((- x) rExpSeq) . (k + 2)) >= 0
t10_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT st A is_all_independent_wrt Prob holds Prob . ((Partial_Intersection (Complement A)) . n) = (Partial_Product (Prob * (Complement A))) . n
t11_bor_cant:: for X being set for A being SetSequence of X holds ( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A )
t12_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma holds ( superior_setsequence A = Union_Shift_Seq A & inferior_setsequence A = Intersect_Shift_Seq A ) by Th11;
t13_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st Partial_Sums (Prob * A) is convergent holds ( Prob . (@lim_sup A) = 0 & lim (Sum_Shift_Seq (Prob,A)) = 0 & Sum_Shift_Seq (Prob,A) is convergent )
t14_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega holds ( ( for X being set for A being SetSequence of X for n being Element of NAT for x being set holds ( ex k being Element of NAT st x in (A ^\ n) . k iff ex k being Element of NAT st ( k >= n & x in A . k ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in @Intersection (Union_Shift_Seq A) iff for m being Element of NAT ex n being Element of NAT st ( n >= m & x in A . n ) ) ) & ( for X being set for A being SetSequence of X for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being set holds ( x in Union (Intersect_Shift_Seq A) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds x in A . k ) ) & ( for A being SetSequence of Sigma for x being Element of Omega holds ( x in Union (Intersect_Shift_Seq (Complement A)) iff ex n being Element of NAT st for k being Element of NAT st k >= n holds not x in A . k ) ) )
t15_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( lim_sup A = @lim_sup A & lim_inf A = @lim_inf A & @lim_inf (Complement A) = (@lim_sup A) ` & (Prob . (@lim_inf (Complement A))) + (Prob . (@lim_sup A)) = 1 & (Prob . (lim_inf (Complement A))) + (Prob . (lim_sup A)) = 1 )
t16_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma holds ( ( Partial_Sums (Prob * A) is convergent implies ( Prob . (lim_sup A) = 0 & Prob . (lim_inf (Complement A)) = 1 ) ) & ( A is_all_independent_wrt Prob & Partial_Sums (Prob * A) is divergent_to+infty implies ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 ) ) )
t17_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st not Partial_Sums (Prob * A) is convergent & A is_all_independent_wrt Prob holds ( Prob . (lim_inf (Complement A)) = 0 & Prob . (lim_sup A) = 1 )
t18_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma st A is_all_independent_wrt Prob holds ( ( Prob . (lim_inf (Complement A)) = 0 or Prob . (lim_inf (Complement A)) = 1 ) & ( Prob . (lim_sup A) = 0 or Prob . (lim_sup A) = 1 ) )
t19_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n1, n being Element of NAT holds (Partial_Sums (Prob * (A ^\ (n1 + 1)))) . n <= ((Partial_Sums (Prob * A)) . ((n1 + 1) + n)) - ((Partial_Sums (Prob * A)) . n1)
t2_bor_cant:: for x being Element of REAL holds 1 + x <= exp_R . x
t20_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Intersect_Shift_Seq (Complement A)) . n) = 1 - (Prob . ((Union_Shift_Seq A) . n))
t21_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds ( ( Complement A is_all_independent_wrt Prob implies Prob . ((Partial_Intersection A) . n) = (Partial_Product (Prob * A)) . n ) & ( A is_all_independent_wrt Prob implies 1 - (Prob . ((Partial_Union A) . n)) = (Partial_Product (Prob * (Complement A))) . n ) )
t3_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (JSum (Prob * A))) . n = exp_R . (- ((Partial_Sums (Prob * A)) . n))
t4_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Product (Prob * (Complement A))) . n <= (Partial_Product (JSum (Prob * A))) . n
t5_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for n, n1, n2 being Element of NAT holds ( ( for A, B being SetSequence of Sigma st n > n1 & B = A * (Special_Function (n1,n2)) holds (Partial_Product (Prob * B)) . n = ((Partial_Product (Prob * A)) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1)) ) & ( for A, B, C being SetSequence of Sigma for e being sequence of NAT st n > n1 & C = A * e & B = C * (Special_Function (n1,n2)) holds (Partial_Intersection B) . n = ((Partial_Intersection C) . n1) /\ ((Partial_Intersection (C ^\ ((n1 + n2) + 1))) . ((n - n1) - 1)) ) )
t6_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n, n1, n2 being Element of NAT st n > n1 & A is_all_independent_wrt Prob holds Prob . (((Partial_Intersection (Complement A)) . n1) /\ ((Partial_Intersection (A ^\ ((n1 + n2) + 1))) . ((n - n1) - 1))) = ((Partial_Product (Prob * (Complement A))) . n1) * ((Partial_Product (Prob * (A ^\ ((n1 + n2) + 1)))) . ((n - n1) - 1))
t7_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma for n being Element of NAT holds (Partial_Intersection (Complement A)) . n = ((Partial_Union A) . n) `
t8_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for Prob being Probability of Sigma for A being SetSequence of Sigma for n being Element of NAT holds Prob . ((Partial_Intersection (Complement A)) . n) = 1 - (Prob . ((Partial_Union A) . n))
t9_bor_cant:: for Omega being non empty set for Sigma being SigmaField of Omega for A being SetSequence of Sigma for n being Element of NAT holds (Intersect_Shift_Seq (Complement A)) . n = ((Union_Shift_Seq A) . n) `
d1_borsuk_1:: for X, Y being non empty TopSpace for F being Function of X,Y holds ( F is continuous iff for W being Point of X for G being a_neighborhood of F . W ex H being a_neighborhood of W st F .: H c= G );
d10_borsuk_1:: for X being non empty TopSpace for b2 being non empty a_partition of the carrier of X holds ( b2 is u.s.c._decomposition of X iff for A being Subset of X st A in b2 holds for V being a_neighborhood of A ex W being Subset of X st ( W is open & A c= W & W c= V & ( for B being Subset of X st B in b2 & B meets W holds B c= W ) ) );
d11_borsuk_1:: for X being TopSpace for IT being SubSpace of X holds ( IT is closed iff for A being Subset of X st A = the carrier of IT holds A is closed );
d12_borsuk_1:: for X being non empty TopSpace for IT being u.s.c._decomposition of X holds ( IT is DECOMPOSITION-like iff for A being Subset of X st A in IT holds A is compact );
d13_borsuk_1:: for b1 being TopStruct holds ( b1 = I[01] iff for P being Subset of (TopSpaceMetr RealSpace) st P = [.0,1.] holds b1 = (TopSpaceMetr RealSpace) | P );
d14_borsuk_1:: 0[01] = 0 ;
d15_borsuk_1:: 1[01] = 1;
d16_borsuk_1:: for A being non empty TopSpace for B being non empty SubSpace of A for F being Function of A,B holds ( F is being_a_retraction iff for W being Point of A st W in the carrier of B holds F . W = W );
d17_borsuk_1:: for X being non empty TopSpace for Y being non empty SubSpace of X holds ( Y is_a_retract_of X iff ex F being continuous Function of X,Y st F is being_a_retraction );
d18_borsuk_1:: for X being non empty TopSpace for Y being non empty SubSpace of X holds ( Y is_an_SDR_of X iff ex H being continuous Function of [:X,I[01]:],X st for A being Point of X holds ( H . [A,0[01]] = A & H . [A,1[01]] in the carrier of Y & ( A in the carrier of Y implies for T being Point of I[01] holds H . [A,T] = A ) ) );
d2_borsuk_1:: for X, Y being TopSpace for b3 being strict TopSpace holds ( b3 = [:X,Y:] iff ( the carrier of b3 = [: the carrier of X, the carrier of Y:] & the topology of b3 = { (union A) where A is Subset-Family of b3 : A c= { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( X1 in the topology of X & Y1 in the topology of Y ) } } ) );
d3_borsuk_1:: for X, Y being TopSpace for A being Subset of [:X,Y:] holds Base-Appr A = { [:X1,Y1:] where X1 is Subset of X, Y1 is Subset of Y : ( [:X1,Y1:] c= A & X1 is open & Y1 is open ) } ;
d4_borsuk_1:: for X, Y being non empty TopSpace holds Pr1 (X,Y) = .: (pr1 ( the carrier of X, the carrier of Y));
d5_borsuk_1:: for X, Y being non empty TopSpace holds Pr2 (X,Y) = .: (pr2 ( the carrier of X, the carrier of Y));
d6_borsuk_1:: for X being 1-sorted holds TrivDecomp X = Class (id the carrier of X);
d7_borsuk_1:: for X being TopSpace for D being a_partition of the carrier of X for b3 being strict TopSpace holds ( b3 = space D iff ( the carrier of b3 = D & the topology of b3 = { A where A is Subset of D : union A in the topology of X } ) );
d8_borsuk_1:: for X being non empty TopSpace for D being non empty a_partition of the carrier of X holds Proj D = proj D;
d9_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X holds TrivExt D = D \/ { {p} where p is Point of XX : not p in the carrier of X } ;
t1_borsuk_1:: for X being TopStruct for Y being SubSpace of X holds the carrier of Y c= the carrier of X
t10_borsuk_1:: for X, Y being non empty TopSpace for XT being Point of [:X,Y:] ex W being Point of X ex T being Point of Y st XT = [W,T]
t11_borsuk_1:: for X, Y being TopSpace for A, B being Subset of [:X,Y:] st A c= B holds Base-Appr A c= Base-Appr B
t12_borsuk_1:: for X, Y being TopSpace for A being Subset of [:X,Y:] holds union (Base-Appr A) c= A
t13_borsuk_1:: for X, Y being TopSpace for A being Subset of [:X,Y:] st A is open holds A = union (Base-Appr A)
t14_borsuk_1:: for X, Y being TopSpace for A being Subset of [:X,Y:] holds Int A = union (Base-Appr A)
t15_borsuk_1:: for X, Y being non empty TopSpace for A being Subset of [:X,Y:] for H being Subset-Family of [:X,Y:] st ( for e being set st e in H holds ( e c= A & ex X1 being Subset of X ex Y1 being Subset of Y st e = [:X1,Y1:] ) ) holds [:(union ((Pr1 (X,Y)) .: H)),(meet ((Pr2 (X,Y)) .: H)):] c= A
t16_borsuk_1:: for X, Y being non empty TopSpace for H being Subset-Family of [:X,Y:] for C being set st C in (Pr1 (X,Y)) .: H holds ex D being Subset of [:X,Y:] st ( D in H & C = (pr1 ( the carrier of X, the carrier of Y)) .: D )
t17_borsuk_1:: for X, Y being non empty TopSpace for H being Subset-Family of [:X,Y:] for C being set st C in (Pr2 (X,Y)) .: H holds ex D being Subset of [:X,Y:] st ( D in H & C = (pr2 ( the carrier of X, the carrier of Y)) .: D )
t18_borsuk_1:: for X, Y being non empty TopSpace for D being Subset of [:X,Y:] st D is open holds for X1 being Subset of X for Y1 being Subset of Y holds ( ( X1 = (pr1 ( the carrier of X, the carrier of Y)) .: D implies X1 is open ) & ( Y1 = (pr2 ( the carrier of X, the carrier of Y)) .: D implies Y1 is open ) )
t19_borsuk_1:: for X, Y being non empty TopSpace for H being Subset-Family of [:X,Y:] st H is open holds ( (Pr1 (X,Y)) .: H is open & (Pr2 (X,Y)) .: H is open )
t2_borsuk_1:: for X, Y being non empty TopSpace for A being continuous Function of X,Y for G being Subset of Y holds A " (Int G) c= Int (A " G)
t20_borsuk_1:: for X, Y being non empty TopSpace for H being Subset-Family of [:X,Y:] st ( (Pr1 (X,Y)) .: H = {} or (Pr2 (X,Y)) .: H = {} ) holds H = {}
t21_borsuk_1:: for X, Y being non empty TopSpace for H being Subset-Family of [:X,Y:] for X1 being Subset of X for Y1 being Subset of Y st H is Cover of [:X1,Y1:] holds ( ( Y1 <> {} implies (Pr1 (X,Y)) .: H is Cover of X1 ) & ( X1 <> {} implies (Pr2 (X,Y)) .: H is Cover of Y1 ) )
t22_borsuk_1:: for X, Y being TopSpace for H being Subset-Family of X for Y being Subset of X st H is Cover of Y holds ex F being Subset-Family of X st ( F c= H & F is Cover of Y & ( for C being set st C in F holds C meets Y ) )
t23_borsuk_1:: for X, Y being non empty TopSpace for F being Subset-Family of X for H being Subset-Family of [:X,Y:] st F is finite & F c= (Pr1 (X,Y)) .: H holds ex G being Subset-Family of [:X,Y:] st ( G c= H & G is finite & F = (Pr1 (X,Y)) .: G )
t24_borsuk_1:: for X, Y being non empty TopSpace for X1 being Subset of X for Y1 being Subset of Y st [:X1,Y1:] <> {} holds ( (Pr1 (X,Y)) . [:X1,Y1:] = X1 & (Pr2 (X,Y)) . [:X1,Y1:] = Y1 ) by EQREL_1:50;
t25_borsuk_1:: for Y, X being non empty TopSpace for t being Point of Y for A being Subset of X st A is compact holds for G being a_neighborhood of [:A,{t}:] ex V being a_neighborhood of A ex W being a_neighborhood of t st [:V,W:] c= G
t26_borsuk_1:: for X being non empty TopSpace for A being Subset of X st A in TrivDecomp X holds ex x being Point of X st A = {x}
t27_borsuk_1:: for X being non empty TopSpace for D being non empty a_partition of the carrier of X for A being Subset of D holds ( union A in the topology of X iff A in the topology of (space D) )
t28_borsuk_1:: for X being non empty TopSpace for D being non empty a_partition of the carrier of X for W being Point of X holds W in (Proj D) . W by EQREL_1:def_9;
t29_borsuk_1:: for X being non empty TopSpace for D being non empty a_partition of the carrier of X for W being Point of (space D) ex W9 being Point of X st (Proj D) . W9 = W
t3_borsuk_1:: for Y, X being non empty TopSpace for W being Point of Y for A being continuous Function of X,Y for G being a_neighborhood of W holds A " G is a_neighborhood of A " {W}
t30_borsuk_1:: for X being non empty TopSpace for D being non empty a_partition of the carrier of X holds rng (Proj D) = the carrier of (space D)
t31_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X for A being Subset of XX holds ( not A in TrivExt D or A in D or ex x being Point of XX st ( not x in [#] X & A = {x} ) )
t32_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X for x being Point of XX st not x in the carrier of X holds {x} in TrivExt D
t33_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X for W being Point of XX st W in the carrier of X holds (Proj (TrivExt D)) . W = (Proj D) . W
t34_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X for W being Point of XX st not W in the carrier of X holds (Proj (TrivExt D)) . W = {W}
t35_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X for W, W9 being Point of XX st not W in the carrier of X & (Proj (TrivExt D)) . W = (Proj (TrivExt D)) . W9 holds W = W9
t36_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X for e being Point of XX st (Proj (TrivExt D)) . e in the carrier of (space D) holds e in the carrier of X
t37_borsuk_1:: for XX being non empty TopSpace for X being non empty SubSpace of XX for D being non empty a_partition of the carrier of X for e being set st e in the carrier of X holds (Proj (TrivExt D)) . e in the carrier of (space D)
t38_borsuk_1:: for X being non empty TopSpace for D being u.s.c._decomposition of X for t being Point of (space D) for G being a_neighborhood of (Proj D) " {t} holds (Proj D) .: G is a_neighborhood of t
t39_borsuk_1:: for X being non empty TopSpace holds TrivDecomp X is u.s.c._decomposition of X
t4_borsuk_1:: for X being non empty TopSpace for A, B being Subset of X for U being a_neighborhood of B st A c= B holds U is a_neighborhood of A
t40_borsuk_1:: the carrier of I[01] = [.0,1.]
t41_borsuk_1:: for XX being non empty TopSpace for X being non empty closed SubSpace of XX for D being DECOMPOSITION of X st X is_a_retract_of XX holds space D is_a_retract_of space (TrivExt D)
t42_borsuk_1:: for XX being non empty TopSpace for X being non empty closed SubSpace of XX for D being DECOMPOSITION of X st X is_an_SDR_of XX holds space D is_an_SDR_of space (TrivExt D)
t43_borsuk_1:: for r being real number holds ( ( 0 <= r & r <= 1 ) iff r in the carrier of I[01] )
t5_borsuk_1:: for X, Y being TopSpace for B being Subset of [:X,Y:] holds ( B is open iff ex A being Subset-Family of [:X,Y:] st ( B = union A & ( for e being set st e in A holds ex X1 being Subset of X ex Y1 being Subset of Y st ( e = [:X1,Y1:] & X1 is open & Y1 is open ) ) ) )
t6_borsuk_1:: for X, Y being TopSpace for V being Subset of X for W being Subset of Y st V is open & W is open holds [:V,W:] is open
t7_borsuk_1:: for X, Y being TopSpace for V being Subset of X for W being Subset of Y holds Int [:V,W:] = [:(Int V),(Int W):]
t8_borsuk_1:: for X, Y being non empty TopSpace for x being Point of X for y being Point of Y for V being a_neighborhood of x for W being a_neighborhood of y holds [:V,W:] is a_neighborhood of [x,y]
t9_borsuk_1:: for X, Y being non empty TopSpace for A being Subset of X for B being Subset of Y for V being a_neighborhood of A for W being a_neighborhood of B holds [:V,W:] is a_neighborhood of [:A,B:]
d1_borsuk_2:: for T being TopStruct for a, b being Point of T holds ( a,b are_connected iff ex f being Function of I[01],T st ( f is continuous & f . 0 = a & f . 1 = b ) );
d2_borsuk_2:: for T being TopStruct for a, b being Point of T st a,b are_connected holds for b4 being Function of I[01],T holds ( b4 is Path of a,b iff ( b4 is continuous & b4 . 0 = a & b4 . 1 = b ) );
d3_borsuk_2:: for T being TopStruct holds ( T is pathwise_connected iff for a, b being Point of T holds a,b are_connected );
d4_borsuk_2:: for T being pathwise_connected TopStruct for a, b being Point of T for b4 being Function of I[01],T holds ( b4 is Path of a,b iff ( b4 is continuous & b4 . 0 = a & b4 . 1 = b ) );
d5_borsuk_2:: for T being non empty TopSpace for a, b, c being Point of T for P being Path of a,b for Q being Path of b,c st a,b are_connected & b,c are_connected holds for b7 being Path of a,c holds ( b7 = P + Q iff for t being Point of I[01] holds ( ( t <= 1 / 2 implies b7 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b7 . t = Q . ((2 * t) - 1) ) ) );
d6_borsuk_2:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b st a,b are_connected holds for b5 being Path of b,a holds ( b5 = - P iff for t being Point of I[01] holds b5 . t = P . (1 - t) );
d7_borsuk_2:: for T being non empty TopStruct for a, b being Point of T for P, Q being Path of a,b holds ( P,Q are_homotopic iff ex f being Function of [:I[01],I[01]:],T st ( f is continuous & ( for t being Point of I[01] holds ( f . (t,0) = P . t & f . (t,1) = Q . t & f . (0,t) = a & f . (1,t) = b ) ) ) );
s1_borsuk_2:: scheme FrCard{ F1() -> non empty set , F2() -> set , F3( set ) -> set , P1[ set ] } : card { F3(w) where w is Element of F1() : ( w in F2() & P1[w] ) } c= card F2()
t1_borsuk_2:: for T1, S, T2, T being non empty TopSpace for f being Function of T1,S for g being Function of T2,S st T1 is SubSpace of T & T2 is SubSpace of T & ([#] T1) \/ ([#] T2) = [#] T & T1 is compact & T2 is compact & T is T_2 & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds f . p = g . p ) holds ex h being Function of T,S st ( h = f +* g & h is continuous )
t10_borsuk_2:: for S1, S2, T1, T2 being non empty TopSpace for f being continuous Function of S1,T1 for g being continuous Function of S2,T2 for P2 being Subset of [:T1,T2:] st P2 is open holds [:f,g:] " P2 is open
t11_borsuk_2:: canceled;
t12_borsuk_2:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b st a,b are_connected holds P,P are_homotopic
t13_borsuk_2:: for G being non empty TopSpace for w1, w2, w3 being Point of G for h1, h2 being Function of I[01],G st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds ex h3 being Function of I[01],G st ( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 & rng h3 c= (rng h1) \/ (rng h2) ) by Lm3;
t14_borsuk_2:: for T being non empty TopSpace for a, b, c being Point of T for G1 being Path of a,b for G2 being Path of b,c st G1 is continuous & G2 is continuous & G1 . 0 = a & G1 . 1 = b & G2 . 0 = b & G2 . 1 = c holds ( G1 + G2 is continuous & (G1 + G2) . 0 = a & (G1 + G2) . 1 = c )
t15_borsuk_2:: for T being non empty TopSpace for a, b being Point of T st ex f being Function of I[01],T st ( f is continuous & f . 0 = a & f . 1 = b ) holds ex g being Function of I[01],T st ( g is continuous & g . 0 = b & g . 1 = a )
t2_borsuk_2:: for S, T being non empty TopSpace for f being Function of S,T st f is being_homeomorphism holds f " is open
t3_borsuk_2:: for T being non empty TopSpace for a being Point of T ex f being Function of I[01],T st ( f is continuous & f . 0 = a & f . 1 = a )
t4_borsuk_2:: canceled;
t5_borsuk_2:: for T being non empty TopSpace for a being Point of T for P being constant Path of a,a holds P = I[01] --> a
t6_borsuk_2:: for T being non empty TopSpace for a being Point of T for P being constant Path of a,a holds P + P = P
t7_borsuk_2:: for T being non empty TopSpace for a being Point of T for P being constant Path of a,a holds - P = P
t8_borsuk_2:: for X, Y being non empty TopSpace for A being Subset-Family of Y for f being Function of X,Y holds f " (union A) = union (f " A)
t9_borsuk_2:: for S1, S2, T1, T2 being non empty TopSpace for f being continuous Function of S1,T1 for g being continuous Function of S2,T2 for P1, P2 being Subset of [:T1,T2:] st P2 in Base-Appr P1 holds [:f,g:] " P2 is open
t1_borsuk_3:: for S, T being TopSpace holds [#] [:S,T:] = [:([#] S),([#] T):] by BORSUK_1:def_2;
t10_borsuk_3:: for X being non empty TopSpace for Y being non empty compact TopSpace for G being open Subset of [:X,Y:] for x being set st [:{x}, the carrier of Y:] c= G holds ex f being ManySortedSet of the carrier of Y st for i being set st i in the carrier of Y holds ex G1 being Subset of X ex H1 being Subset of Y st ( f . i = [G1,H1] & [x,i] in [:G1,H1:] & G1 is open & H1 is open & [:G1,H1:] c= G )
t11_borsuk_3:: for X being non empty TopSpace for Y being non empty compact TopSpace for G being open Subset of [:Y,X:] for x being set st [:([#] Y),{x}:] c= G holds ex R being open Subset of X st ( x in R & R c= { y where y is Point of X : [:([#] Y),{y}:] c= G } )
t12_borsuk_3:: for X being non empty TopSpace for Y being non empty compact TopSpace for G being open Subset of [:Y,X:] holds { x where x is Point of X : [:([#] Y),{x}:] c= G } in the topology of X
t13_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X holds [:(X | {x}),Y:],Y are_homeomorphic
t14_borsuk_3:: for S, T being non empty TopSpace st S,T are_homeomorphic & S is compact holds T is compact
t15_borsuk_3:: for X, Y being TopSpace for XV being SubSpace of X holds [:Y,XV:] is SubSpace of [:Y,X:]
t16_borsuk_3:: for X being non empty TopSpace for Y being non empty compact TopSpace for x being Point of X for Z being Subset of [:Y,X:] st Z = [:([#] Y),{x}:] holds Z is compact
t17_borsuk_3:: for X, Y being non empty compact TopSpace for R being Subset-Family of X st R = { Q where Q is open Subset of X : [:([#] Y),Q:] c= union (Base-Appr ([#] [:Y,X:])) } holds ( R is open & R is Cover of [#] X )
t18_borsuk_3:: for X, Y being non empty compact TopSpace for R being Subset-Family of X for F being Subset-Family of [:Y,X:] st F is Cover of [:Y,X:] & F is open & R = { Q where Q is open Subset of X : ex FQ being Subset-Family of [:Y,X:] st ( FQ c= F & FQ is finite & [:([#] Y),Q:] c= union FQ ) } holds ( R is open & R is Cover of X )
t19_borsuk_3:: for X, Y being non empty compact TopSpace for R being Subset-Family of X for F being Subset-Family of [:Y,X:] st F is Cover of [:Y,X:] & F is open & R = { Q where Q is open Subset of X : ex FQ being Subset-Family of [:Y,X:] st ( FQ c= F & FQ is finite & [:([#] Y),Q:] c= union FQ ) } holds ex C being Subset-Family of X st ( C c= R & C is finite & C is Cover of X )
t2_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X holds Y --> x is continuous Function of Y,(X | {x})
t20_borsuk_3:: for X, Y being non empty compact TopSpace for F being Subset-Family of [:Y,X:] st F is Cover of [:Y,X:] & F is open holds ex G being Subset-Family of [:Y,X:] st ( G c= F & G is Cover of [:Y,X:] & G is finite )
t21_borsuk_3:: for X, Y being TopSpace for XV being SubSpace of X for YV being SubSpace of Y holds [:XV,YV:] is SubSpace of [:X,Y:]
t22_borsuk_3:: for X, Y being TopSpace for Z being Subset of [:Y,X:] for V being Subset of X for W being Subset of Y st Z = [:W,V:] holds TopStruct(# the carrier of [:(Y | W),(X | V):], the topology of [:(Y | W),(X | V):] #) = TopStruct(# the carrier of ([:Y,X:] | Z), the topology of ([:Y,X:] | Z) #)
t23_borsuk_3:: for T1, T2 being TopSpace for S1 being Subset of T1 for S2 being Subset of T2 st S1 is compact & S2 is compact holds [:S1,S2:] is compact Subset of [:T1,T2:]
t3_borsuk_3:: for S, T, V being non empty TopSpace st S,T are_homeomorphic & T,V are_homeomorphic holds S,V are_homeomorphic
t4_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X for f being Function of [:Y,(X | {x}):],Y st f = pr1 ( the carrier of Y,{x}) holds f is one-to-one
t5_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X for f being Function of [:(X | {x}),Y:],Y st f = pr2 ({x}, the carrier of Y) holds f is one-to-one
t6_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X for f being Function of [:Y,(X | {x}):],Y st f = pr1 ( the carrier of Y,{x}) holds f " = <:(id Y),(Y --> x):>
t7_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X for f being Function of [:(X | {x}),Y:],Y st f = pr2 ({x}, the carrier of Y) holds f " = <:(Y --> x),(id Y):>
t8_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X for f being Function of [:Y,(X | {x}):],Y st f = pr1 ( the carrier of Y,{x}) holds f is being_homeomorphism
t9_borsuk_3:: for X, Y being non empty TopSpace for x being Point of X for f being Function of [:(X | {x}),Y:],Y st f = pr2 ({x}, the carrier of Y) holds f is being_homeomorphism
d1_borsuk_4:: for b1 being strict SubSpace of I[01] holds ( b1 = I(01) iff the carrier of b1 = ].0,1.[ );
t1_borsuk_4:: for X being non empty set for A being non empty Subset of X st A is trivial holds ex x being Element of X st A = {x} by SUBSET_1:47;
t10_borsuk_4:: for A being Subset of I[01] for a, b being real number st a < b & A = [.a,b.[ holds [.a,b.] c= the carrier of I[01]
t11_borsuk_4:: for a, b being real number st a <> b holds Cl ].a,b.] = [.a,b.]
t12_borsuk_4:: for a, b being real number st a <> b holds Cl [.a,b.[ = [.a,b.]
t13_borsuk_4:: for A being Subset of I[01] for a, b being real number st a < b & A = ].a,b.[ holds Cl A = [.a,b.]
t14_borsuk_4:: for A being Subset of I[01] for a, b being real number st a < b & A = ].a,b.] holds Cl A = [.a,b.]
t15_borsuk_4:: for A being Subset of I[01] for a, b being real number st a < b & A = [.a,b.[ holds Cl A = [.a,b.]
t16_borsuk_4:: for A being Subset of I[01] for a, b being real number st a <= b & A = [.a,b.] holds ( 0 <= a & b <= 1 )
t17_borsuk_4:: for A, B being Subset of I[01] for a, b, c being real number st a < b & b < c & A = [.a,b.[ & B = ].b,c.] holds A,B are_separated
t18_borsuk_4:: for p1, p2 being Point of I[01] holds [.p1,p2.] is Subset of I[01] by BORSUK_1:40, XXREAL_2:def_12;
t19_borsuk_4:: for a, b being Point of I[01] holds ].a,b.[ is Subset of I[01]
t2_borsuk_4:: for X being non trivial set for p being set ex q being Element of X st q <> p
t20_borsuk_4:: for p being real number holds {p} is non empty closed_interval Subset of REAL
t21_borsuk_4:: for A being non empty connected Subset of I[01] for a, b, c being Point of I[01] st a <= b & b <= c & a in A & c in A holds b in A
t22_borsuk_4:: for A being non empty connected Subset of I[01] for a, b being real number st a in A & b in A holds [.a,b.] c= A
t23_borsuk_4:: for a, b being real number for A being Subset of I[01] st A = [.a,b.] holds A is closed
t24_borsuk_4:: for p1, p2 being Point of I[01] st p1 <= p2 holds [.p1,p2.] is non empty connected compact Subset of I[01]
t25_borsuk_4:: for X being Subset of I[01] for X9 being Subset of REAL st X9 = X holds ( X9 is bounded_above & X9 is bounded_below )
t26_borsuk_4:: for X being Subset of I[01] for X9 being Subset of REAL for x being real number st x in X9 & X9 = X holds ( lower_bound X9 <= x & x <= upper_bound X9 )
t27_borsuk_4:: for A being Subset of REAL for B being Subset of I[01] st A = B holds ( A is closed iff B is closed )
t28_borsuk_4:: for C being non empty closed_interval Subset of REAL holds lower_bound C <= upper_bound C
t29_borsuk_4:: for C being non empty connected compact Subset of I[01] for C9 being Subset of REAL st C = C9 & [.(lower_bound C9),(upper_bound C9).] c= C9 holds [.(lower_bound C9),(upper_bound C9).] = C9
t3_borsuk_4:: for T being non trivial set for X being non trivial Subset of T for p being set ex q being Element of T st ( q in X & q <> p )
t30_borsuk_4:: for C being non empty connected compact Subset of I[01] holds C is non empty closed_interval Subset of REAL
t31_borsuk_4:: for C being non empty connected compact Subset of I[01] ex p1, p2 being Point of I[01] st ( p1 <= p2 & C = [.p1,p2.] )
t32_borsuk_4:: for A being Subset of I[01] st A = the carrier of I(01) holds I(01) = I[01] | A by PRE_TOPC:8, TSEP_1:5;
t33_borsuk_4:: the carrier of I(01) = the carrier of I[01] \ {0,1}
t34_borsuk_4:: I(01) is open ;
t35_borsuk_4:: for r being real number holds ( r in the carrier of I(01) iff ( 0 < r & r < 1 ) )
t36_borsuk_4:: for a, b being Point of I[01] st a < b & b <> 1 holds ].a,b.] is non empty Subset of I(01)
t37_borsuk_4:: for a, b being Point of I[01] st a < b & a <> 0 holds [.a,b.[ is non empty Subset of I(01)
t38_borsuk_4:: for D being Simple_closed_curve holds (TOP-REAL 2) | R^2-unit_square,(TOP-REAL 2) | D are_homeomorphic
t39_borsuk_4:: for n being Element of NAT for D being non empty Subset of (TOP-REAL n) for p1, p2 being Point of (TOP-REAL n) st D is_an_arc_of p1,p2 holds I(01) ,(TOP-REAL n) | (D \ {p1,p2}) are_homeomorphic
t4_borsuk_4:: for f, g being Function for a being set st f is one-to-one & g is one-to-one & (dom f) /\ (dom g) = {a} & (rng f) /\ (rng g) = {(f . a)} holds f +* g is one-to-one
t40_borsuk_4:: for n being Element of NAT for D being Subset of (TOP-REAL n) for p1, p2 being Point of (TOP-REAL n) st D is_an_arc_of p1,p2 holds I[01] ,(TOP-REAL n) | D are_homeomorphic
t41_borsuk_4:: for n being Element of NAT for p1, p2 being Point of (TOP-REAL n) st p1 <> p2 holds I[01] ,(TOP-REAL n) | (LSeg (p1,p2)) are_homeomorphic by Th40, TOPREAL1:9;
t42_borsuk_4:: for E being Subset of I(01) st ex p1, p2 being Point of I[01] st ( p1 < p2 & E = [.p1,p2.] ) holds I[01] ,I(01) | E are_homeomorphic
t43_borsuk_4:: for n being Element of NAT for A being Subset of (TOP-REAL n) for p, q being Point of (TOP-REAL n) for a, b being Point of I[01] st A is_an_arc_of p,q & a < b holds ex E being non empty Subset of I[01] ex f being Function of (I[01] | E),((TOP-REAL n) | A) st ( E = [.a,b.] & f is being_homeomorphism & f . a = p & f . b = q )
t44_borsuk_4:: for A being TopSpace for B being non empty TopSpace for f being Function of A,B for C being TopSpace for X being Subset of A st f is continuous & C is SubSpace of B holds for h being Function of (A | X),C st h = f | X holds h is continuous
t45_borsuk_4:: for X being Subset of I[01] for a, b being Point of I[01] st X = ].a,b.[ holds X is open
t46_borsuk_4:: for X being Subset of I(01) for a, b being Point of I[01] st X = ].a,b.[ holds X is open
t47_borsuk_4:: for X being Subset of I(01) for a being Point of I[01] st X = ].0,a.] holds X is closed
t48_borsuk_4:: for X being Subset of I(01) for a being Point of I[01] st X = [.a,1.[ holds X is closed
t49_borsuk_4:: for n being Element of NAT for A being Subset of (TOP-REAL n) for p, q being Point of (TOP-REAL n) for a, b being Point of I[01] st A is_an_arc_of p,q & a < b & b <> 1 holds ex E being non empty Subset of I(01) ex f being Function of (I(01) | E),((TOP-REAL n) | (A \ {p})) st ( E = ].a,b.] & f is being_homeomorphism & f . b = q )
t5_borsuk_4:: for f, g being Function for a being set st f is one-to-one & g is one-to-one & (dom f) /\ (dom g) = {a} & (rng f) /\ (rng g) = {(f . a)} & f . a = g . a holds (f +* g) " = (f ") +* (g ")
t50_borsuk_4:: for n being Element of NAT for A being Subset of (TOP-REAL n) for p, q being Point of (TOP-REAL n) for a, b being Point of I[01] st A is_an_arc_of p,q & a < b & a <> 0 holds ex E being non empty Subset of I(01) ex f being Function of (I(01) | E),((TOP-REAL n) | (A \ {q})) st ( E = [.a,b.[ & f is being_homeomorphism & f . a = p )
t51_borsuk_4:: for n being Element of NAT for A, B being Subset of (TOP-REAL n) for p, q being Point of (TOP-REAL n) st A is_an_arc_of p,q & B is_an_arc_of q,p & A /\ B = {p,q} & p <> q holds I(01) ,(TOP-REAL n) | ((A \ {p}) \/ (B \ {p})) are_homeomorphic
t52_borsuk_4:: for D being Simple_closed_curve for p being Point of (TOP-REAL 2) st p in D holds (TOP-REAL 2) | (D \ {p}), I(01) are_homeomorphic
t53_borsuk_4:: for D being Simple_closed_curve for p, q being Point of (TOP-REAL 2) st p in D & q in D holds (TOP-REAL 2) | (D \ {p}),(TOP-REAL 2) | (D \ {q}) are_homeomorphic
t54_borsuk_4:: for n being Element of NAT for C being non empty Subset of (TOP-REAL n) for E being Subset of I(01) st ex p1, p2 being Point of I[01] st ( p1 < p2 & E = [.p1,p2.] ) & I(01) | E,(TOP-REAL n) | C are_homeomorphic holds ex s1, s2 being Point of (TOP-REAL n) st C is_an_arc_of s1,s2
t55_borsuk_4:: for Dp being non empty Subset of (TOP-REAL 2) for f being Function of ((TOP-REAL 2) | Dp),I(01) for C being non empty Subset of (TOP-REAL 2) st f is being_homeomorphism & C c= Dp & ex p1, p2 being Point of I[01] st ( p1 < p2 & f .: C = [.p1,p2.] ) holds ex s1, s2 being Point of (TOP-REAL 2) st C is_an_arc_of s1,s2
t56_borsuk_4:: for D being Simple_closed_curve for C being non empty connected compact Subset of (TOP-REAL 2) holds ( not C c= D or C = D or ex p1, p2 being Point of (TOP-REAL 2) st C is_an_arc_of p1,p2 or ex p being Point of (TOP-REAL 2) st C = {p} )
t57_borsuk_4:: for C being non empty compact Subset of I[01] st C c= ].0,1.[ holds ex D being non empty closed_interval Subset of REAL st ( C c= D & D c= ].0,1.[ & lower_bound C = lower_bound D & upper_bound C = upper_bound D )
t58_borsuk_4:: for C being non empty compact Subset of I[01] st C c= ].0,1.[ holds ex p1, p2 being Point of I[01] st ( p1 <= p2 & C c= [.p1,p2.] & [.p1,p2.] c= ].0,1.[ )
t59_borsuk_4:: for D being Simple_closed_curve for C being closed Subset of (TOP-REAL 2) st C c< D holds ex p1, p2 being Point of (TOP-REAL 2) ex B being Subset of (TOP-REAL 2) st ( B is_an_arc_of p1,p2 & C c= B & B c= D )
t6_borsuk_4:: for n being Element of NAT for A being Subset of (TOP-REAL n) for p, q being Point of (TOP-REAL n) st A is_an_arc_of p,q holds not A \ {p} is empty
t7_borsuk_4:: for s1, s3, s4, l being real number st s1 <= s3 & s1 < s4 & 0 <= l & l <= 1 holds s1 <= ((1 - l) * s3) + (l * s4)
t8_borsuk_4:: for A being Subset of I[01] for a, b being real number st a < b & A = ].a,b.[ holds [.a,b.] c= the carrier of I[01]
t9_borsuk_4:: for A being Subset of I[01] for a, b being real number st a < b & A = ].a,b.] holds [.a,b.] c= the carrier of I[01]
d1_borsuk_5:: IRRAT = REAL \ RAT;
d2_borsuk_5:: for a, b being real number holds RAT (a,b) = RAT /\ ].a,b.[;
d3_borsuk_5:: for a, b being real number holds IRRAT (a,b) = IRRAT /\ ].a,b.[;
t1_borsuk_5:: canceled;
t10_borsuk_5:: for A, B being Subset of R^1 for a, b, c, d being real number st a < b & b <= c & c < d & A = [.a,b.[ & B = ].c,d.] holds A,B are_separated
t11_borsuk_5:: for a, b, c being real number st a <= c & c <= b holds [.a,b.] \/ [.c,+infty.[ = [.a,+infty.[
t12_borsuk_5:: for a, b, c being real number st a <= c & c <= b holds ].-infty,c.] \/ [.a,b.] = ].-infty,b.]
t13_borsuk_5:: for A being Subset of R^1 for p being Point of RealSpace holds ( p in Cl A iff for r being real number st r > 0 holds Ball (p,r) meets A ) by GOBOARD6:92, TOPMETR:def_6;
t14_borsuk_5:: for p, q being Element of RealSpace st q >= p holds dist (p,q) = q - p
t15_borsuk_5:: for A being Subset of R^1 st A = RAT holds Cl A = the carrier of R^1
t16_borsuk_5:: for A being Subset of R^1 for a, b being real number st A = ].a,b.[ & a <> b holds Cl A = [.a,b.]
t17_borsuk_5:: for x being real number holds ( x is irrational iff x in IRRAT )
t18_borsuk_5:: for a being rational number for b being real irrational number holds a + b is irrational ;
t19_borsuk_5:: for a being real irrational number holds - a is irrational ;
t2_borsuk_5:: for x1, x2, x3, x4, x5, x6 being set holds {x1,x2,x3,x4,x5,x6} = {x1,x3,x6} \/ {x2,x4,x5}
t20_borsuk_5:: for a being rational number for b being real irrational number holds a - b is irrational ;
t21_borsuk_5:: for a being rational number for b being real irrational number holds b - a is irrational ;
t22_borsuk_5:: for a being rational number for b being real irrational number st a <> 0 holds a * b is irrational
t23_borsuk_5:: for a being rational number for b being real irrational number st a <> 0 holds b / a is irrational
t24_borsuk_5:: for a being rational number for b being real irrational number st a <> 0 holds a / b is irrational
t25_borsuk_5:: for r being real irrational number holds frac r is irrational ;
t26_borsuk_5:: for a, b being real number st a < b holds ex p1, p2 being rational number st ( a < p1 & p1 < p2 & p2 < b )
t27_borsuk_5:: for a, b being real number st a < b holds ex p being real irrational number st ( a < p & p < b )
t28_borsuk_5:: for A being Subset of R^1 st A = IRRAT holds Cl A = the carrier of R^1
t29_borsuk_5:: for a, b, c being real number holds ( c in RAT (a,b) iff ( c is rational & a < c & c < b ) )
t3_borsuk_5:: for x1, x2, x3, x4, x5, x6 being set st x1,x2,x3,x4,x5,x6 are_mutually_different holds card {x1,x2,x3,x4,x5,x6} = 6
t30_borsuk_5:: for a, b, c being real number holds ( c in IRRAT (a,b) iff ( c is irrational & a < c & c < b ) )
t31_borsuk_5:: for A being Subset of R^1 for a, b being real number st a < b & A = RAT (a,b) holds Cl A = [.a,b.]
t32_borsuk_5:: for A being Subset of R^1 for a, b being real number st a < b & A = IRRAT (a,b) holds Cl A = [.a,b.]
t33_borsuk_5:: for T being connected TopSpace for A being open closed Subset of T holds ( A = {} or A = [#] T )
t34_borsuk_5:: for A being Subset of R^1 st A is closed & A is open & not A = {} holds A = REAL
t35_borsuk_5:: for A being Subset of R^1 for a, b being real number st A = [.a,b.[ & a <> b holds Cl A = [.a,b.]
t36_borsuk_5:: for A being Subset of R^1 for a, b being real number st A = ].a,b.] & a <> b holds Cl A = [.a,b.]
t37_borsuk_5:: for A being Subset of R^1 for a, b, c being real number st A = [.a,b.[ \/ ].b,c.] & a < b & b < c holds Cl A = [.a,c.]
t38_borsuk_5:: for A being Subset of R^1 for a being real number st A = {a} holds Cl A = {a}
t39_borsuk_5:: for A being Subset of REAL for B being Subset of R^1 st A = B holds ( A is open iff B is open )
t4_borsuk_5:: for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds card {x1,x2,x3,x4,x5,x6,x7} = 7
t40_borsuk_5:: for A being Subset of R^1 for a, b being ext-real number st A = ].a,b.[ holds A is open
t41_borsuk_5:: for A being Subset of R^1 for a being real number st A = ].-infty,a.] holds A is closed
t42_borsuk_5:: for A being Subset of R^1 for a being real number st A = [.a,+infty.[ holds A is closed
t43_borsuk_5:: for a being real number holds [.a,+infty.[ = {a} \/ ].a,+infty.[
t44_borsuk_5:: for a being real number holds ].-infty,a.] = {a} \/ ].-infty,a.[
t45_borsuk_5:: for a being real number holds ].a,+infty.[ <> REAL
t46_borsuk_5:: for a being real number holds [.a,+infty.[ <> REAL
t47_borsuk_5:: for a being real number holds ].-infty,a.] <> REAL
t48_borsuk_5:: for a being real number holds ].-infty,a.[ <> REAL
t49_borsuk_5:: for A being Subset of R^1 for a being real number st A = ].a,+infty.[ holds Cl A = [.a,+infty.[
t5_borsuk_5:: for x1, x2, x3, x4, x5, x6 being set st {x1,x2,x3} misses {x4,x5,x6} holds ( x1 <> x4 & x1 <> x5 & x1 <> x6 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6 )
t50_borsuk_5:: for a being real number holds Cl ].a,+infty.[ = [.a,+infty.[
t51_borsuk_5:: for A being Subset of R^1 for a being real number st A = ].-infty,a.[ holds Cl A = ].-infty,a.]
t52_borsuk_5:: for a being real number holds Cl ].-infty,a.[ = ].-infty,a.]
t53_borsuk_5:: for A, B being Subset of R^1 for b being real number st A = ].-infty,b.[ & B = ].b,+infty.[ holds A,B are_separated
t54_borsuk_5:: for A being Subset of R^1 for a, b being real number st a < b & A = [.a,b.[ \/ ].b,+infty.[ holds Cl A = [.a,+infty.[
t55_borsuk_5:: for A being Subset of R^1 for a, b being real number st a < b & A = ].a,b.[ \/ ].b,+infty.[ holds Cl A = [.a,+infty.[
t56_borsuk_5:: for A being Subset of R^1 for a, b, c being real number st a < b & b < c & A = ((RAT (a,b)) \/ ].b,c.[) \/ ].c,+infty.[ holds Cl A = [.a,+infty.[
t57_borsuk_5:: for a, b being real number holds IRRAT (a,b) misses RAT (a,b)
t58_borsuk_5:: for a, b being real number holds REAL \ (RAT (a,b)) = (].-infty,a.] \/ (IRRAT (a,b))) \/ [.b,+infty.[
t59_borsuk_5:: for a, b being real number st a < b holds [.a,+infty.[ \ (].a,b.[ \/ ].b,+infty.[) = {a} \/ {b}
t6_borsuk_5:: for x1, x2, x3, x4, x5, x6 being set st x1,x2,x3 are_mutually_different & x4,x5,x6 are_mutually_different & {x1,x2,x3} misses {x4,x5,x6} holds x1,x2,x3,x4,x5,x6 are_mutually_different
t60_borsuk_5:: for A being Subset of R^1 st A = ((RAT (2,4)) \/ ].4,5.[) \/ ].5,+infty.[ holds A ` = ((].-infty,2.] \/ (IRRAT (2,4))) \/ {4}) \/ {5}
t61_borsuk_5:: for A being Subset of R^1 for a being real number st A = {a} holds A ` = ].-infty,a.[ \/ ].a,+infty.[ by TOPMETR:17, XXREAL_1:389;
t62_borsuk_5:: (].-infty,1.[ \/ ].1,+infty.[) /\ (((].-infty,2.] \/ (IRRAT (2,4))) \/ {4}) \/ {5}) = (((].-infty,1.[ \/ ].1,2.]) \/ (IRRAT (2,4))) \/ {4}) \/ {5}
t63_borsuk_5:: for A being Subset of R^1 for a, b being real number st a <= b & A = {a} \/ [.b,+infty.[ holds A ` = ].-infty,a.[ \/ ].a,b.[
t64_borsuk_5:: for A being Subset of R^1 for a, b being real number st a < b & A = ].-infty,a.[ \/ ].a,b.[ holds Cl A = ].-infty,b.]
t65_borsuk_5:: for A being Subset of R^1 for a, b being real number st a < b & A = ].-infty,a.[ \/ ].a,b.] holds Cl A = ].-infty,b.]
t66_borsuk_5:: for A being Subset of R^1 for a, b, c being real number st a < b & b < c & A = ((].-infty,a.[ \/ ].a,b.]) \/ (IRRAT (b,c))) \/ {c} holds Cl A = ].-infty,c.]
t67_borsuk_5:: for A being Subset of R^1 for a, b, c, d being real number st a < b & b < c & A = (((].-infty,a.[ \/ ].a,b.]) \/ (IRRAT (b,c))) \/ {c}) \/ {d} holds Cl A = ].-infty,c.] \/ {d}
t68_borsuk_5:: for A being Subset of R^1 for a, b being real number st a <= b & A = ].-infty,a.] \/ {b} holds A ` = ].a,b.[ \/ ].b,+infty.[
t69_borsuk_5:: for a, b being real number holds [.a,+infty.[ \/ {b} <> REAL
t7_borsuk_5:: for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6 are_mutually_different & {x1,x2,x3,x4,x5,x6} misses {x7} holds x1,x2,x3,x4,x5,x6,x7 are_mutually_different
t70_borsuk_5:: for a, b being real number holds ].-infty,a.] \/ {b} <> REAL
t71_borsuk_5:: for TS being TopStruct for A, B being Subset of TS st A <> B holds A ` <> B `
t72_borsuk_5:: for A being Subset of R^1 st REAL = A ` holds A = {}
t73_borsuk_5:: for X being compact Subset of R^1 for X9 being Subset of REAL st X9 = X holds ( X9 is bounded_above & X9 is bounded_below )
t74_borsuk_5:: for X being compact Subset of R^1 for X9 being Subset of REAL for x being real number st x in X9 & X9 = X holds ( lower_bound X9 <= x & x <= upper_bound X9 )
t75_borsuk_5:: for C being non empty connected compact Subset of R^1 for C9 being Subset of REAL st C = C9 & [.(lower_bound C9),(upper_bound C9).] c= C9 holds [.(lower_bound C9),(upper_bound C9).] = C9
t76_borsuk_5:: for A being connected Subset of R^1 for a, b, c being real number st a <= b & b <= c & a in A & c in A holds b in A
t77_borsuk_5:: for A being connected Subset of R^1 for a, b being real number st a in A & b in A holds [.a,b.] c= A
t78_borsuk_5:: for X being non empty connected compact Subset of R^1 holds X is non empty closed_interval Subset of REAL
t79_borsuk_5:: for A being non empty connected compact Subset of R^1 ex a, b being real number st ( a <= b & A = [.a,b.] )
t8_borsuk_5:: for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds x7,x1,x2,x3,x4,x5,x6 are_mutually_different
t9_borsuk_5:: for x1, x2, x3, x4, x5, x6, x7 being set st x1,x2,x3,x4,x5,x6,x7 are_mutually_different holds x1,x2,x5,x3,x6,x7,x4 are_mutually_different
d1_borsuk_6:: for a, b, c, d being real number holds L[01] (a,b,c,d) = (L[01] (((#) (c,d)),((c,d) (#)))) * (P[01] (a,b,((#) (0,1)),((0,1) (#))));
d10_borsuk_6:: for b1 being Subset of [:I[01],I[01]:] holds ( b1 = LowerRightUnitTriangle iff for x being set holds ( x in b1 iff ex a, b being Point of I[01] st ( x = [a,b] & b <= (2 * a) - 1 ) ) );
d11_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P, Q being Path of a,b st P,Q are_homotopic holds for b6 being Function of [:I[01],I[01]:],T holds ( b6 is Homotopy of P,Q iff ( b6 is continuous & ( for t being Point of I[01] holds ( b6 . (t,0) = P . t & b6 . (t,1) = Q . t & b6 . (0,t) = a & b6 . (1,t) = b ) ) ) );
d2_borsuk_6:: for T being non empty pathwise_connected TopSpace for a, b, c being Point of T for P being Path of a,b for Q being Path of b,c for b7 being Path of a,c holds ( b7 = P + Q iff for t being Point of I[01] holds ( ( t <= 1 / 2 implies b7 . t = P . (2 * t) ) & ( 1 / 2 <= t implies b7 . t = Q . ((2 * t) - 1) ) ) );
d3_borsuk_6:: for T being non empty pathwise_connected TopSpace for a, b being Point of T for P being Path of a,b for b5 being Path of b,a holds ( b5 = - P iff for t being Point of I[01] holds b5 . t = P . (1 - t) );
d4_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b for f being continuous Function of I[01],I[01] st f . 0 = 0 & f . 1 = 1 & a,b are_connected holds RePar (P,f) = P * f;
d5_borsuk_6:: for b1 being Function of I[01],I[01] holds ( b1 = 1RP iff for t being Point of I[01] holds ( ( t <= 1 / 2 implies b1 . t = 2 * t ) & ( t > 1 / 2 implies b1 . t = 1 ) ) );
d6_borsuk_6:: for b1 being Function of I[01],I[01] holds ( b1 = 2RP iff for t being Point of I[01] holds ( ( t <= 1 / 2 implies b1 . t = 0 ) & ( t > 1 / 2 implies b1 . t = (2 * t) - 1 ) ) );
d7_borsuk_6:: for b1 being Function of I[01],I[01] holds ( b1 = 3RP iff for x being Point of I[01] holds ( ( x <= 1 / 2 implies b1 . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies b1 . x = x - (1 / 4) ) & ( x > 3 / 4 implies b1 . x = (2 * x) - 1 ) ) );
d8_borsuk_6:: for b1 being Subset of [:I[01],I[01]:] holds ( b1 = LowerLeftUnitTriangle iff for x being set holds ( x in b1 iff ex a, b being Point of I[01] st ( x = [a,b] & b <= 1 - (2 * a) ) ) );
d9_borsuk_6:: for b1 being Subset of [:I[01],I[01]:] holds ( b1 = UpperUnitTriangle iff for x being set holds ( x in b1 iff ex a, b being Point of I[01] st ( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) );
s1_borsuk_6:: scheme ExFunc3CondD{ F1() -> non empty set , P1[ set ], P2[ set ], P3[ set ], F2( set ) -> set , F3( set ) -> set , F4( set ) -> set } : ex f being Function st ( dom f = F1() & ( for c being Element of F1() holds ( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) ) ) ) provided A1: for c being Element of F1() holds ( ( P1[c] implies not P2[c] ) & ( P1[c] implies not P3[c] ) & ( P2[c] implies not P3[c] ) ) and A2: for c being Element of F1() holds ( P1[c] or P2[c] or P3[c] )
t1_borsuk_6:: the carrier of [:I[01],I[01]:] = [:[.0,1.],[.0,1.]:] by BORSUK_1:40, BORSUK_1:def_2;
t10_borsuk_6:: for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1)) - 1 } & T = { p where p is Point of (TOP-REAL 2) : p `2 <= p `1 } holds (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
t11_borsuk_6:: for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 >= (2 * (p `1)) - 1 } & T = { p where p is Point of (TOP-REAL 2) : p `2 >= p `1 } holds (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
t12_borsuk_6:: for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 >= 1 - (2 * (p `1)) } & T = { p where p is Point of (TOP-REAL 2) : p `2 >= - (p `1) } holds (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T
t13_borsuk_6:: for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 <= 1 - (2 * (p `1)) } & T = { p where p is Point of (TOP-REAL 2) : p `2 <= - (p `1) } holds (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T
t14_borsuk_6:: for T being non empty 1-sorted holds ( T is real-membered iff for x being Element of T holds x is real )
t15_borsuk_6:: { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1)) - 1 } is closed Subset of (TOP-REAL 2)
t16_borsuk_6:: { p where p is Point of (TOP-REAL 2) : p `2 >= (2 * (p `1)) - 1 } is closed Subset of (TOP-REAL 2)
t17_borsuk_6:: { p where p is Point of (TOP-REAL 2) : p `2 <= 1 - (2 * (p `1)) } is closed Subset of (TOP-REAL 2)
t18_borsuk_6:: { p where p is Point of (TOP-REAL 2) : p `2 >= 1 - (2 * (p `1)) } is closed Subset of (TOP-REAL 2)
t19_borsuk_6:: { p where p is Point of (TOP-REAL 2) : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } is closed Subset of (TOP-REAL 2)
t2_borsuk_6:: for a, b, x being real number st a <= x & x <= b holds (x - a) / (b - a) in the carrier of (Closed-Interval-TSpace (0,1))
t20_borsuk_6:: ex f being Function of [:R^1,R^1:],(TOP-REAL 2) st for x, y being Real holds f . [x,y] = <*x,y*>
t21_borsuk_6:: { p where p is Point of [:R^1,R^1:] : p `2 <= 1 - (2 * (p `1)) } is closed Subset of [:R^1,R^1:]
t22_borsuk_6:: { p where p is Point of [:R^1,R^1:] : p `2 <= (2 * (p `1)) - 1 } is closed Subset of [:R^1,R^1:]
t23_borsuk_6:: { p where p is Point of [:R^1,R^1:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } is closed Subset of [:R^1,R^1:]
t24_borsuk_6:: { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } is non empty closed Subset of [:I[01],I[01]:]
t25_borsuk_6:: { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } is non empty closed Subset of [:I[01],I[01]:]
t26_borsuk_6:: { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } is non empty closed Subset of [:I[01],I[01]:]
t27_borsuk_6:: for S, T being non empty TopSpace for p being Point of [:S,T:] holds ( p `1 is Point of S & p `2 is Point of T )
t28_borsuk_6:: for A, B being Subset of [:I[01],I[01]:] st A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] holds ([#] ([:I[01],I[01]:] | A)) \/ ([#] ([:I[01],I[01]:] | B)) = [#] [:I[01],I[01]:]
t29_borsuk_6:: for A, B being Subset of [:I[01],I[01]:] st A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] holds ([#] ([:I[01],I[01]:] | A)) /\ ([#] ([:I[01],I[01]:] | B)) = [:{(1 / 2)},[.0,1.]:]
t3_borsuk_6:: for x being Point of I[01] st x <= 1 / 2 holds 2 * x is Point of I[01]
t30_borsuk_6:: for T being TopStruct holds {} is empty compact Subset of T
t31_borsuk_6:: for T being TopStruct for a, b being real number st a > b holds [.a,b.] is empty compact Subset of T
t32_borsuk_6:: for a, b, c, d being Point of I[01] holds [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:]
t33_borsuk_6:: for a, b, c, d being real number st a < b & c < d holds ( (L[01] (a,b,c,d)) . a = c & (L[01] (a,b,c,d)) . b = d )
t34_borsuk_6:: for a, b, c, d being real number st a < b & c <= d holds L[01] (a,b,c,d) is continuous Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (c,d))
t35_borsuk_6:: for a, b, c, d being real number st a < b & c <= d holds for x being real number st a <= x & x <= b holds (L[01] (a,b,c,d)) . x = (((d - c) / (b - a)) * (x - a)) + c
t36_borsuk_6:: for f1, f2 being Function of [:I[01],I[01]:],I[01] st f1 is continuous & f2 is continuous & ( for p being Point of [:I[01],I[01]:] holds (f1 . p) * (f2 . p) is Point of I[01] ) holds ex g being Function of [:I[01],I[01]:],I[01] st ( ( for p being Point of [:I[01],I[01]:] for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 * r2 ) & g is continuous )
t37_borsuk_6:: for f1, f2 being Function of [:I[01],I[01]:],I[01] st f1 is continuous & f2 is continuous & ( for p being Point of [:I[01],I[01]:] holds (f1 . p) + (f2 . p) is Point of I[01] ) holds ex g being Function of [:I[01],I[01]:],I[01] st ( ( for p being Point of [:I[01],I[01]:] for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 + r2 ) & g is continuous )
t38_borsuk_6:: for f1, f2 being Function of [:I[01],I[01]:],I[01] st f1 is continuous & f2 is continuous & ( for p being Point of [:I[01],I[01]:] holds (f1 . p) - (f2 . p) is Point of I[01] ) holds ex g being Function of [:I[01],I[01]:],I[01] st ( ( for p being Point of [:I[01],I[01]:] for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds g . p = r1 - r2 ) & g is continuous )
t39_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b st P is continuous holds P * (L[01] (((0,1) (#)),((#) (0,1)))) is continuous Function of I[01],T
t4_borsuk_6:: for x being Point of I[01] st x >= 1 / 2 holds (2 * x) - 1 is Point of I[01]
t40_borsuk_6:: for X being non empty TopStruct for a, b being Point of X for P being Path of a,b st P . 0 = a & P . 1 = b holds ( (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 0 = b & (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 1 = a )
t41_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b st P is continuous & P . 0 = a & P . 1 = b holds ( - P is continuous & (- P) . 0 = b & (- P) . 1 = a )
t42_borsuk_6:: for T being non empty TopSpace for a, b, c being Point of T st a,b are_connected & b,c are_connected holds a,c are_connected
t43_borsuk_6:: for T being non empty TopSpace for a, b being Point of T st a,b are_connected holds for A being Path of a,b holds A = - (- A)
t44_borsuk_6:: for T being non empty pathwise_connected TopSpace for a, b being Point of T for A being Path of a,b holds A = - (- A)
t45_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b for f being continuous Function of I[01],I[01] st f . 0 = 0 & f . 1 = 1 & a,b are_connected holds RePar (P,f),P are_homotopic
t46_borsuk_6:: for T being non empty pathwise_connected TopSpace for a, b being Point of T for P being Path of a,b for f being continuous Function of I[01],I[01] st f . 0 = 0 & f . 1 = 1 holds RePar (P,f),P are_homotopic
t47_borsuk_6:: ( 1RP . 0 = 0 & 1RP . 1 = 1 )
t48_borsuk_6:: ( 2RP . 0 = 0 & 2RP . 1 = 1 )
t49_borsuk_6:: ( 3RP . 0 = 0 & 3RP . 1 = 1 )
t5_borsuk_6:: for p, q being Point of I[01] holds p * q is Point of I[01]
t50_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b for Q being constant Path of b,b st a,b are_connected holds RePar (P,1RP) = P + Q
t51_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b for Q being constant Path of a,a st a,b are_connected holds RePar (P,2RP) = Q + P
t52_borsuk_6:: for T being non empty TopSpace for a, b, c, d being Point of T for P being Path of a,b for Q being Path of b,c for R being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds RePar (((P + Q) + R),3RP) = P + (Q + R)
t53_borsuk_6:: IAA = { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) }
t54_borsuk_6:: IBB = { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) }
t55_borsuk_6:: ICC = { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 }
t56_borsuk_6:: (IAA \/ IBB) \/ ICC = [:[.0,1.],[.0,1.]:]
t57_borsuk_6:: IAA /\ IBB = { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) }
t58_borsuk_6:: ICC /\ IBB = { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 }
t59_borsuk_6:: for x being Point of [:I[01],I[01]:] st x in IAA holds x `1 <= 1 / 2
t6_borsuk_6:: for x being Point of I[01] holds (1 / 2) * x is Point of I[01]
t60_borsuk_6:: for x being Point of [:I[01],I[01]:] st x in ICC holds x `1 >= 1 / 2
t61_borsuk_6:: for x being Point of I[01] holds [0,x] in IAA
t62_borsuk_6:: for s being set st [0,s] in IBB holds s = 1
t63_borsuk_6:: for s being set st [s,1] in ICC holds s = 1
t64_borsuk_6:: [0,1] in IBB
t65_borsuk_6:: for x being Point of I[01] holds [x,1] in IBB
t66_borsuk_6:: ( [(1 / 2),0] in ICC & [1,1] in ICC )
t67_borsuk_6:: [(1 / 2),0] in IBB
t68_borsuk_6:: for x being Point of I[01] holds [1,x] in ICC
t69_borsuk_6:: for x being Point of I[01] st x >= 1 / 2 holds [x,0] in ICC
t7_borsuk_6:: for x being Point of I[01] st x >= 1 / 2 holds x - (1 / 4) is Point of I[01]
t70_borsuk_6:: for x being Point of I[01] st x <= 1 / 2 holds [x,0] in IAA
t71_borsuk_6:: for x being Point of I[01] st x < 1 / 2 holds ( not [x,0] in IBB & not [x,0] in ICC )
t72_borsuk_6:: IAA /\ ICC = {[(1 / 2),0]}
t73_borsuk_6:: for T being non empty TopSpace for a, b, c, d being Point of T for P being Path of a,b for Q being Path of b,c for R being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds (P + Q) + R,P + (Q + R) are_homotopic
t74_borsuk_6:: for X being non empty pathwise_connected TopSpace for a1, b1, c1, d1 being Point of X for P being Path of a1,b1 for Q being Path of b1,c1 for R being Path of c1,d1 holds (P + Q) + R,P + (Q + R) are_homotopic
t75_borsuk_6:: for T being non empty TopSpace for a, b, c being Point of T for P1, P2 being Path of a,b for Q1, Q2 being Path of b,c st a,b are_connected & b,c are_connected & P1,P2 are_homotopic & Q1,Q2 are_homotopic holds P1 + Q1,P2 + Q2 are_homotopic
t76_borsuk_6:: for X being non empty pathwise_connected TopSpace for a1, b1, c1 being Point of X for P1, P2 being Path of a1,b1 for Q1, Q2 being Path of b1,c1 st P1,P2 are_homotopic & Q1,Q2 are_homotopic holds P1 + Q1,P2 + Q2 are_homotopic
t77_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P, Q being Path of a,b st a,b are_connected & P,Q are_homotopic holds - P, - Q are_homotopic
t78_borsuk_6:: for X being non empty pathwise_connected TopSpace for a1, b1 being Point of X for P, Q being Path of a1,b1 st P,Q are_homotopic holds - P, - Q are_homotopic
t79_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P, Q, R being Path of a,b st P,Q are_homotopic & Q,R are_homotopic holds P,R are_homotopic
t8_borsuk_6:: id I[01] is Path of 0[01] , 1[01]
t80_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b for Q being constant Path of b,b st a,b are_connected holds P + Q,P are_homotopic
t81_borsuk_6:: for X being non empty pathwise_connected TopSpace for a1, b1 being Point of X for P being Path of a1,b1 for Q being constant Path of b1,b1 holds P + Q,P are_homotopic
t82_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b for Q being constant Path of a,a st a,b are_connected holds Q + P,P are_homotopic
t83_borsuk_6:: for X being non empty pathwise_connected TopSpace for a1, b1 being Point of X for P being Path of a1,b1 for Q being constant Path of a1,a1 holds Q + P,P are_homotopic
t84_borsuk_6:: for T being non empty TopSpace for a, b being Point of T for P being Path of a,b for Q being constant Path of a,a st a,b are_connected holds P + (- P),Q are_homotopic
t85_borsuk_6:: for X being non empty pathwise_connected TopSpace for a1, b1 being Point of X for P being Path of a1,b1 for Q being constant Path of a1,a1 holds P + (- P),Q are_homotopic
t86_borsuk_6:: for T being non empty TopSpace for b, a being Point of T for P being Path of b,a for Q being constant Path of a,a st b,a are_connected holds (- P) + P,Q are_homotopic
t87_borsuk_6:: for X being non empty pathwise_connected TopSpace for b1, a1 being Point of X for P being Path of b1,a1 for Q being constant Path of a1,a1 holds (- P) + P,Q are_homotopic
t88_borsuk_6:: for T being non empty TopSpace for a being Point of T for P, Q being constant Path of a,a holds P,Q are_homotopic
t9_borsuk_6:: for a, b, c, d being Point of I[01] st a <= b & c <= d holds [:[.a,b.],[.c,d.]:] is non empty compact Subset of [:I[01],I[01]:]
d1_borsuk_7:: for a, b, c, x, y, z being set holds (a,b,c) --> (x,y,z) = ((a,b) --> (x,y)) +* (c .--> z);
d10_borsuk_7:: for n being non zero Nat for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),1)),(TOP-REAL n) for b3 being Function of (Tunit_circle (n + 1)),(Tunit_circle n) holds ( b3 = Sn1->Sn f iff for x, y being Point of (Tcircle ((0. (TOP-REAL (n + 1))),1)) st y = - x holds b3 . x = Rn->S1 ((f . x) - (f . y)) );
d11_borsuk_7:: for x0, y0 being Point of (Tunit_circle 2) for xt being set for f being Path of x0,y0 st xt in CircleMap " {x0} holds for b5 being Function of I[01],R^1 holds ( b5 = liftPath (f,xt) iff ( b5 . 0 = xt & f = CircleMap * b5 & b5 is continuous & ( for f1 being Function of I[01],R^1 st f1 is continuous & f = CircleMap * f1 & f1 . 0 = xt holds b5 = f1 ) ) );
d12_borsuk_7:: for n being Nat for p, x, y being Point of (TOP-REAL n) for r being real number holds ( x,y are_antipodals_of p,r iff ( x is Point of (Tcircle (p,r)) & y is Point of (Tcircle (p,r)) & p in LSeg (x,y) ) );
d13_borsuk_7:: for n being Nat for p, x, y being Point of (TOP-REAL n) for r being real number for f being Function holds ( x,y are_antipodals_of p,r,f iff ( x,y are_antipodals_of p,r & x in dom f & y in dom f & f . x = f . y ) );
d14_borsuk_7:: for m, n being Nat for p being Point of (TOP-REAL m) for r being real number for f being Function of (Tcircle (p,r)),(TOP-REAL n) holds ( f is with_antipodals iff ex x, y being Point of (TOP-REAL m) st x,y are_antipodals_of p,r,f );
d2_borsuk_7:: for f being Function holds ( f is odd iff for x, y being complex-valued Function st x in dom f & - x in dom f & y = f . x holds f . (- x) = - y );
d3_borsuk_7:: c[100] = |[1,0,0]|;
d4_borsuk_7:: c[-100] = |[(- 1),0,0]|;
d5_borsuk_7:: for r being real non negative number for s being Real holds RotateCircle (r,s) = (Rotate s) | (Tcircle ((0. (TOP-REAL 2)),r));
d6_borsuk_7:: for n being non empty Nat for p being Point of (TOP-REAL n) for r being real non negative number for b4 being Function of (Tunit_circle n),(Tcircle (p,r)) holds ( b4 = CircleIso (p,r) iff for a being Point of (Tunit_circle n) for b being Point of (TOP-REAL n) st a = b holds b4 . a = (r * b) + p );
d7_borsuk_7:: for b1 being Function of R^1,(Tunit_circle 3) holds ( b1 = SphereMap iff for x being real number holds b1 . x = |[(cos ((2 * PI) * x)),(sin ((2 * PI) * x)),0]| );
d8_borsuk_7:: for r being real number for b2 being Function of I[01],(Tunit_circle 3) holds ( b2 = eLoop r iff for x being Point of I[01] holds b2 . x = |[(cos (((2 * PI) * r) * x)),(sin (((2 * PI) * r) * x)),0]| );
d9_borsuk_7:: for n being Nat for p being Point of (TOP-REAL n) st p <> 0. (TOP-REAL n) holds Rn->S1 p = p (/) |.p.|;
t1_borsuk_7:: for r, s being real number st 0 <= r & 0 <= s & r ^2 = s ^2 holds r = s
t10_borsuk_7:: for r, s being real number st cos r = cos s holds ex i being Integer st ( r = s + ((2 * PI) * i) or r = (- s) + ((2 * PI) * i) )
t11_borsuk_7:: for r, s being real number st sin r = sin s & cos r = cos s holds ex i being Integer st r = s + ((2 * PI) * i)
t12_borsuk_7:: for i being Integer for c1, c2 being complex number st |.c1.| = |.c2.| & Arg c1 = (Arg c2) + ((2 * PI) * i) holds c1 = c2
t13_borsuk_7:: for f being complex-valued FinSequence holds len (- f) = len f
t14_borsuk_7:: for n being Nat holds - (0* n) = 0* n
t15_borsuk_7:: for n being Nat for f being complex-valued Function st f <> 0* n holds - f <> 0* n
t16_borsuk_7:: for r1, r2, r3 being real number holds sqr <*r1,r2,r3*> = <*(r1 ^2),(r2 ^2),(r3 ^2)*>
t17_borsuk_7:: for r1, r2, r3 being real number holds Sum (sqr <*r1,r2,r3*>) = ((r1 ^2) + (r2 ^2)) + (r3 ^2)
t18_borsuk_7:: for c being complex number for f being complex-valued FinSequence holds (c (#) f) ^2 = (c ^2) (#) (f ^2)
t19_borsuk_7:: for c being complex number for f being complex-valued FinSequence holds (f (/) c) ^2 = (f ^2) (/) (c ^2)
t2_borsuk_7:: for r, s being real number st frac r >= frac s holds frac (r - s) = (frac r) - (frac s)
t20_borsuk_7:: for f being real-valued FinSequence st Sum f <> 0 holds Sum (f (/) (Sum f)) = 1
t21_borsuk_7:: for a, b, x, y, z being set for c being complex number holds dom ((a,b,c) --> (x,y,z)) = {a,b,c} by BVFUNC14:11;
t22_borsuk_7:: for a, b, x, y, z being set for c being complex number holds rng ((a,b,c) --> (x,y,z)) c= {x,y,z}
t23_borsuk_7:: for a, x, y, z being set holds (a,a,a) --> (x,y,z) = a .--> z
t24_borsuk_7:: for a, b, x, y, z being set holds (a,a,b) --> (x,y,z) = (a,b) --> (y,z) by FUNCT_4:81;
t25_borsuk_7:: for a, b, x, y, z being set st a <> b holds (a,b,a) --> (x,y,z) = (a,b) --> (z,y)
t26_borsuk_7:: for a, b, x, y, z being set holds (a,b,b) --> (x,y,z) = (a,b) --> (x,z)
t27_borsuk_7:: for a, b, x, y, z being set for c being complex number st a <> b & a <> c holds ((a,b,c) --> (x,y,z)) . a = x by BVFUNC14:13;
t28_borsuk_7:: for a, b, x, y, z being set for c being complex number st a,b,c are_mutually_different holds ( ((a,b,c) --> (x,y,z)) . a = x & ((a,b,c) --> (x,y,z)) . b = y & ((a,b,c) --> (x,y,z)) . c = z )
t29_borsuk_7:: for a, b, x, y, z being set for c being complex number for f being Function st dom f = {a,b,c} & f . a = x & f . b = y & f . c = z holds f = (a,b,c) --> (x,y,z)
t3_borsuk_7:: for r, s being real number st frac r < frac s holds frac (r - s) = ((frac r) - (frac s)) + 1
t30_borsuk_7:: for a, b being set for c being complex number holds <*a,b,c*> = (1,2,3) --> (a,b,c)
t31_borsuk_7:: for a, b, x, y, z being set for c being complex number st a,b,c are_mutually_different holds product ((a,b,c) --> ({x},{y},{z})) = {((a,b,c) --> (x,y,z))}
t32_borsuk_7:: for a, b being set for c being complex number for A, B, C, D, E, F being set st A c= B & C c= D & E c= F holds product ((a,b,c) --> (A,C,E)) c= product ((a,b,c) --> (B,D,F))
t33_borsuk_7:: for a, b, x, X, y, Y, z, Z being set for c being complex number st a,b,c are_mutually_different & x in X & y in Y & z in Z holds (a,b,c) --> (x,y,z) in product ((a,b,c) --> (X,Y,Z))
t34_borsuk_7:: for p being Point of (TOP-REAL 3) holds sqr p = <*((p `1) ^2),((p `2) ^2),((p `3) ^2)*>
t35_borsuk_7:: for p being Point of (TOP-REAL 3) holds Sum (sqr p) = (((p `1) ^2) + ((p `2) ^2)) + ((p `3) ^2)
t36_borsuk_7:: for r being real number for S being Subset of R^1 st S = RAT holds RAT /\ ].-infty,r.[ is open Subset of (R^1 | S)
t37_borsuk_7:: for r being real number for S being Subset of R^1 st S = RAT holds RAT /\ ].r,+infty.[ is open Subset of (R^1 | S)
t38_borsuk_7:: for S being Subset of R^1 st S = RAT holds for T being connected TopSpace for f being Function of T,(R^1 | S) st f is continuous holds f is constant
t39_borsuk_7:: for a, b being real number for f being continuous Function of (Closed-Interval-TSpace (a,b)),R^1 for g being PartFunc of REAL,REAL st a <= b & f = g holds g is continuous
t4_borsuk_7:: for r, s being real number ex i being Integer st ( frac (r - s) = ((frac r) - (frac s)) + i & ( i = 0 or i = 1 ) )
t40_borsuk_7:: - c[100] = c[-100]
t41_borsuk_7:: - c[-100] = c[100] by Th40;
t42_borsuk_7:: c[100] - c[-100] = |[2,0,0]|
t43_borsuk_7:: for p being Point of (TOP-REAL 2) holds ( p `1 = |.p.| * (cos (Arg p)) & p `2 = |.p.| * (sin (Arg p)) )
t44_borsuk_7:: for p being Point of (TOP-REAL 2) holds p = cpx2euc ((|.p.| * (cos (Arg p))) + ((|.p.| * (sin (Arg p))) * *))
t45_borsuk_7:: for i being Integer for p1, p2 being Point of (TOP-REAL 2) st |.p1.| = |.p2.| & Arg p1 = (Arg p2) + ((2 * PI) * i) holds p1 = p2
t46_borsuk_7:: for r being real number for p being Point of (TOP-REAL 2) st p = CircleMap . r holds Arg p = (2 * PI) * (frac r)
t47_borsuk_7:: for p1, p2 being Point of (TOP-REAL 3) for u1, u2 being Point of (Euclid 3) st u1 = p1 & u2 = p2 holds (Pitag_dist 3) . (u1,u2) = sqrt (((((p1 `1) - (p2 `1)) ^2) + (((p1 `2) - (p2 `2)) ^2)) + (((p1 `3) - (p2 `3)) ^2))
t48_borsuk_7:: for r being real number for p being Point of (TOP-REAL 3) for e being Point of (Euclid 3) st p = e & p `3 = 0 holds product ((1,2,3) --> (].((p `1) - (r / (sqrt 2))),((p `1) + (r / (sqrt 2))).[,].((p `2) - (r / (sqrt 2))),((p `2) + (r / (sqrt 2))).[,{0})) c= Ball (e,r)
t49_borsuk_7:: for i being Integer for c being complex number for s being Real holds Rotate (c,s) = Rotate (c,(s + ((2 * PI) * i)))
t5_borsuk_7:: for r being real number holds ( not sin r = 0 or r = (2 * PI) * [\(r / (2 * PI))/] or r = PI + ((2 * PI) * [\(r / (2 * PI))/]) )
t50_borsuk_7:: for i being Integer for s being Real holds Rotate s = Rotate (s + ((2 * PI) * i))
t51_borsuk_7:: for s being Real for p being Point of (TOP-REAL 2) holds |.((Rotate s) . p).| = |.p.|
t52_borsuk_7:: for s being Real for p being Point of (TOP-REAL 2) holds Arg ((Rotate s) . p) = Arg (Rotate ((euc2cpx p),s))
t53_borsuk_7:: for s being Real for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds ex i being Integer st Arg ((Rotate s) . p) = (s + (Arg p)) + ((2 * PI) * i)
t54_borsuk_7:: for s being Real holds (Rotate s) . (0. (TOP-REAL 2)) = 0. (TOP-REAL 2)
t55_borsuk_7:: for s being Real for p being Point of (TOP-REAL 2) st (Rotate s) . p = 0. (TOP-REAL 2) holds p = 0. (TOP-REAL 2)
t56_borsuk_7:: for s being Real for p being Point of (TOP-REAL 2) holds (Rotate s) . ((Rotate (- s)) . p) = p
t57_borsuk_7:: for s being Real holds (Rotate s) * (Rotate (- s)) = id (TOP-REAL 2)
t58_borsuk_7:: for r being real number for s being Real for p being Point of (TOP-REAL 2) holds ( p in Sphere ((0. (TOP-REAL 2)),r) iff (Rotate s) . p in Sphere ((0. (TOP-REAL 2)),r) )
t59_borsuk_7:: for r being real non negative number for s being Real holds (Rotate s) .: (Sphere ((0. (TOP-REAL 2)),r)) = Sphere ((0. (TOP-REAL 2)),r)
t6_borsuk_7:: for r being real number holds ( not cos r = 0 or r = (PI / 2) + ((2 * PI) * [\(r / (2 * PI))/]) or r = ((3 * PI) / 2) + ((2 * PI) * [\(r / (2 * PI))/]) )
t60_borsuk_7:: for r2, r1 being real number for p being Point of (TOP-REAL 2) st p = CircleMap . r2 holds (RotateCircle (1,(- (Arg p)))) . (CircleMap . r1) = CircleMap . (r1 - r2)
t61_borsuk_7:: for i being Integer holds SphereMap . i = c[100]
t62_borsuk_7:: for r being real number holds eLoop r = SphereMap * (ExtendInt r)
t63_borsuk_7:: for n being Nat for p being Point of (TOP-REAL n) st p <> 0. (TOP-REAL n) holds |.(p (/) |.p.|).| = 1
t64_borsuk_7:: for n being non empty Nat for r being real non negative number for x being Point of (TOP-REAL n) st x is Point of (Tcircle ((0. (TOP-REAL n)),r)) holds x, - x are_antipodals_of 0. (TOP-REAL n),r
t65_borsuk_7:: for n being non empty Nat for p, x, y, x1, y1 being Point of (TOP-REAL n) for r being real positive number st x,y are_antipodals_of 0. (TOP-REAL n),1 & x1 = (CircleIso (p,r)) . x & y1 = (CircleIso (p,r)) . y holds x1,y1 are_antipodals_of p,r
t66_borsuk_7:: for n being non zero Nat for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),1)),(TOP-REAL n) for x being Point of (Tcircle ((0. (TOP-REAL (n + 1))),1)) st f is without_antipodals holds (f . x) - (f . (- x)) <> 0. (TOP-REAL n)
t67_borsuk_7:: for n being non zero Nat for f being Function of (Tcircle ((0. (TOP-REAL (n + 1))),1)),(TOP-REAL n) st f is without_antipodals holds Sn1->Sn f is odd
t68_borsuk_7:: for n being non zero Nat for f, g, B1 being Function of (Tcircle ((0. (TOP-REAL (n + 1))),1)),(TOP-REAL n) st g = f (-) & B1 = f <--> g & f is without_antipodals holds Sn1->Sn f = B1 ((n NormF) * B1)
t7_borsuk_7:: for r being real number st sin r = 0 holds ex i being Integer st r = PI * i
t8_borsuk_7:: for r being real number st cos r = 0 holds ex i being Integer st r = (PI / 2) + (PI * i)
t9_borsuk_7:: for r, s being real number st sin r = sin s holds ex i being Integer st ( r = s + ((2 * PI) * i) or r = (PI - s) + ((2 * PI) * i) )
d1_brouwer:: for S, T being non empty TopSpace holds DiffElems (S,T) = { [s,t] where s is Point of S, t is Point of T : s <> t } ;
d2_brouwer:: for n being Element of NAT for x being Point of (TOP-REAL n) for r being real number holds Tdisk (x,r) = (TOP-REAL n) | (cl_Ball (x,r));
d3_brouwer:: for n being non empty Element of NAT for o, s, t being Point of (TOP-REAL n) for r being non negative real number st s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t holds for b6 being Point of (TOP-REAL n) holds ( b6 = HC (s,t,o,r) iff ( b6 in (halfline (s,t)) /\ (Sphere (o,r)) & b6 <> s ) );
d4_brouwer:: for n being non empty Element of NAT for o being Point of (TOP-REAL n) for r being non negative real number for x being Point of (Tdisk (o,r)) for f being Function of (Tdisk (o,r)),(Tdisk (o,r)) st not x is_a_fixpoint_of f holds for b6 being Point of (Tcircle (o,r)) holds ( b6 = HC (x,f) iff ex y, z being Point of (TOP-REAL n) st ( y = x & z = f . x & b6 = HC (z,y,o,r) ) );
d5_brouwer:: for n being non empty Element of NAT for r being non negative real number for o being Point of (TOP-REAL n) for f being Function of (Tdisk (o,r)),(Tdisk (o,r)) for b5 being Function of (Tdisk (o,r)),(Tcircle (o,r)) holds ( b5 = BR-map f iff for x being Point of (Tdisk (o,r)) holds b5 . x = HC (x,f) );
t1_brouwer:: for S, T being non empty TopSpace for x being set holds ( x in DiffElems (S,T) iff ex s being Point of S ex t being Point of T st ( x = [s,t] & s <> t ) ) ;
t10_brouwer:: for r being positive real number for o being Point of (TOP-REAL 2) for Y being non empty SubSpace of Tdisk (o,r) st Y = Tcircle (o,r) holds not Y is_a_retract_of Tdisk (o,r)
t11_brouwer:: for r being non negative real number for n being non empty Element of NAT for o being Point of (TOP-REAL n) for x being Point of (Tdisk (o,r)) for f being Function of (Tdisk (o,r)),(Tdisk (o,r)) st not x is_a_fixpoint_of f & x is Point of (Tcircle (o,r)) holds (BR-map f) . x = x
t12_brouwer:: for r being non negative real number for n being non empty Element of NAT for o being Point of (TOP-REAL n) for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) st f is without_fixpoints holds (BR-map f) | (Sphere (o,r)) = id (Tcircle (o,r))
t13_brouwer:: for r being positive real number for o being Point of (TOP-REAL 2) for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) st f is without_fixpoints holds BR-map f is continuous
t14_brouwer:: for r being non negative real number for o being Point of (TOP-REAL 2) for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) holds f is with_fixpoint
t15_brouwer:: for r being non negative real number for o being Point of (TOP-REAL 2) for f being continuous Function of (Tdisk (o,r)),(Tdisk (o,r)) ex x being Point of (Tdisk (o,r)) st f . x = x
t2_brouwer:: for n being Element of NAT for x being Point of (TOP-REAL n) holds cl_Ball (x,0) = {x}
t3_brouwer:: for n being Element of NAT for r being real number for x being Point of (TOP-REAL n) holds the carrier of (Tdisk (x,r)) = cl_Ball (x,r)
t4_brouwer:: for n being Element of NAT for r being non negative real number for s, t, x being Point of (TOP-REAL n) st s <> t & s is Point of (Tdisk (x,r)) & s is not Point of (Tcircle (x,r)) holds ex e being Point of (Tcircle (x,r)) st {e} = (halfline (s,t)) /\ (Sphere (x,r))
t5_brouwer:: for n being Element of NAT for r being non negative real number for s, t, x being Point of (TOP-REAL n) st s <> t & s in the carrier of (Tcircle (x,r)) & t is Point of (Tdisk (x,r)) holds ex e being Point of (Tcircle (x,r)) st ( e <> s & {s,e} = (halfline (s,t)) /\ (Sphere (x,r)) )
t6_brouwer:: for r being non negative real number for n being non empty Element of NAT for s, o, t being Point of (TOP-REAL n) st s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t holds HC (s,t,o,r) is Point of (Tcircle (o,r))
t7_brouwer:: for a being real number for r being non negative real number for n being non empty Element of NAT for s, t, o being Point of (TOP-REAL n) for S, T, O being Element of REAL n st S = s & T = t & O = o & s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t & a = ((- |((t - s),(s - o))|) + (sqrt ((|((t - s),(s - o))| ^2) - ((Sum (sqr (T - S))) * ((Sum (sqr (S - O))) - (r ^2)))))) / (Sum (sqr (T - S))) holds HC (s,t,o,r) = ((1 - a) * s) + (a * t)
t8_brouwer:: for a being real number for r being non negative real number for r1, r2, s1, s2 being real number for s, t, o being Point of (TOP-REAL 2) st s is Point of (Tdisk (o,r)) & t is Point of (Tdisk (o,r)) & s <> t & r1 = (t `1) - (s `1) & r2 = (t `2) - (s `2) & s1 = (s `1) - (o `1) & s2 = (s `2) - (o `2) & a = ((- ((s1 * r1) + (s2 * r2))) + (sqrt ((((s1 * r1) + (s2 * r2)) ^2) - (((r1 ^2) + (r2 ^2)) * (((s1 ^2) + (s2 ^2)) - (r ^2)))))) / ((r1 ^2) + (r2 ^2)) holds HC (s,t,o,r) = |[((s `1) + (a * r1)),((s `2) + (a * r2))]|
t9_brouwer:: for r being non negative real number for n being non empty Element of NAT for o being Point of (TOP-REAL n) for x being Point of (Tdisk (o,r)) for f being Function of (Tdisk (o,r)),(Tdisk (o,r)) st not x is_a_fixpoint_of f & x is Point of (Tcircle (o,r)) holds HC (x,f) = x
t1_brouwer2:: for n being Nat for p, q being Point of (TOP-REAL n) for r being real number holds ((1 - r) * p) + (r * q) = p + (r * (q - p))
t2_brouwer2:: for n being Nat for u, p, q, w being Point of (TOP-REAL n) st u in halfline (p,q) & w in halfline (p,q) & |.(u - p).| = |.(w - p).| holds u = w
t3_brouwer2:: for n being Nat for p, q being Point of (TOP-REAL n) for S being Subset of (TOP-REAL n) st p in S & p <> q & S /\ (halfline (p,q)) is bounded holds ex w being Point of (TOP-REAL n) st ( w in (Fr S) /\ (halfline (p,q)) & ( for u being Point of (TOP-REAL n) st u in S /\ (halfline (p,q)) holds |.(p - u).| <= |.(p - w).| ) & ( for r being real number st r > 0 holds ex u being Point of (TOP-REAL n) st ( u in S /\ (halfline (p,q)) & |.(w - u).| < r ) ) )
t4_brouwer2:: for n being Nat for p, q being Point of (TOP-REAL n) for A being convex Subset of (TOP-REAL n) st A is closed & p in Int A & p <> q & A /\ (halfline (p,q)) is bounded holds ex u being Point of (TOP-REAL n) st (Fr A) /\ (halfline (p,q)) = {u}
t5_brouwer2:: for n being Nat for p being Point of (TOP-REAL n) for r being real number st r > 0 holds Fr (cl_Ball (p,r)) = Sphere (p,r)
t6_brouwer2:: for n being Element of NAT for A being convex Subset of (TOP-REAL n) st A is compact & not A is boundary holds ex h being Function of ((TOP-REAL n) | A),(Tdisk ((0. (TOP-REAL n)),1)) st ( h is being_homeomorphism & h .: (Fr A) = Sphere ((0. (TOP-REAL n)),1) )
t7_brouwer2:: for n being Nat for A, B being convex Subset of (TOP-REAL n) st A is compact & not A is boundary & B is compact & not B is boundary holds ex h being Function of ((TOP-REAL n) | A),((TOP-REAL n) | B) st ( h is being_homeomorphism & h .: (Fr A) = Fr B )
t8_brouwer2:: for n being Nat for A being convex Subset of (TOP-REAL n) st A is compact & not A is boundary holds for h being continuous Function of ((TOP-REAL n) | A),((TOP-REAL n) | A) holds h is with_fixpoint
t9_brouwer2:: for n being Nat for A being non empty convex Subset of (TOP-REAL n) st A is compact & not A is boundary holds for FR being non empty SubSpace of (TOP-REAL n) | A st [#] FR = Fr A holds not FR is_a_retract_of (TOP-REAL n) | A
d1_bspace:: for S being 1-sorted holds <*> S = <*> ([#] S);
d2_bspace:: Z_2 = INT.Ring 2;
d3_bspace:: for X, x being set holds ( ( x in X implies X @ x = 1. Z_2 ) & ( not x in X implies X @ x = 0. Z_2 ) );
d4_bspace:: for X being set for a being Element of Z_2 for c being Subset of X holds ( ( a = 1. Z_2 implies a \*\ c = c ) & ( a = 0. Z_2 implies a \*\ c = {} X ) );
d5_bspace:: for X being set for b2 being BinOp of (bool X) holds ( b2 = bspace-sum X iff for c, d being Subset of X holds b2 . (c,d) = c \+\ d );
d6_bspace:: for X being set for b2 being Function of [: the carrier of Z_2,(bool X):],(bool X) holds ( b2 = bspace-scalar-mult X iff for a being Element of Z_2 for c being Subset of X holds b2 . (a,c) = a \*\ c );
d7_bspace:: for X being set holds bspace X = VectSpStr(# (bool X),(bspace-sum X),({} X),(bspace-scalar-mult X) #);
d8_bspace:: for X being set holds singletons X = { f where f is Subset of X : f is 1 -element } ;
d9_bspace:: for X being non empty set for s being FinSequence of (bspace X) for x being Element of X for b4 being FinSequence of Z_2 holds ( b4 = s @ x iff ( len b4 = len s & ( for j being Nat st 1 <= j & j <= len s holds b4 . j = (s . j) @ x ) ) );
s1_bspace:: scheme IndSeqS{ F1() -> 1-sorted , P1[ set ] } : for p being FinSequence of F1() holds P1[p] provided A1: P1[ <*> F1()] and A2: for p being FinSequence of F1() for x being Element of F1() st P1[p] holds P1[p ^ <*x*>]
t1_bspace:: for S being 1-sorted for i being Element of NAT for p being FinSequence of S st i in dom p holds p . i in S
t10_bspace:: for X, x being set holds ( X @ x = 0. Z_2 iff not x in X ) by Def3;
t11_bspace:: for X, x being set holds ( X @ x <> 0. Z_2 iff X @ x = 1. Z_2 ) by Th5, Th6, CARD_1:50, TARSKI:def_2;
t12_bspace:: for X, x, y being set holds ( X @ x = X @ y iff ( x in X iff y in X ) )
t13_bspace:: for X, Y, x being set holds ( X @ x = Y @ x iff ( x in X iff x in Y ) )
t14_bspace:: for x being set holds {} @ x = 0. Z_2 by Def3;
t15_bspace:: for X being set for u, v being Subset of X for x being Element of X holds (u \+\ v) @ x = (u @ x) + (v @ x)
t16_bspace:: for X, Y being set holds ( X = Y iff for x being set holds X @ x = Y @ x )
t17_bspace:: for X being set for a being Element of Z_2 for c, d being Subset of X holds a \*\ (c \+\ d) = (a \*\ c) \+\ (a \*\ d)
t18_bspace:: for X being set for a, b being Element of Z_2 for c being Subset of X holds (a + b) \*\ c = (a \*\ c) \+\ (b \*\ c)
t19_bspace:: for X being set for c being Subset of X holds (1. Z_2) \*\ c = c by Def4;
t2_bspace:: for S being 1-sorted for p being FinSequence st ( for i being Nat st i in dom p holds p . i in S ) holds p is FinSequence of S
t20_bspace:: for X being set for a, b being Element of Z_2 for c being Subset of X holds a \*\ (b \*\ c) = (a * b) \*\ c
t21_bspace:: for X being set holds bspace X is Abelian
t22_bspace:: for X being set holds bspace X is add-associative
t23_bspace:: for X being set holds bspace X is right_zeroed
t24_bspace:: for X being set holds bspace X is right_complementable
t25_bspace:: for X being set for a being Element of Z_2 for x, y being Element of (bspace X) holds a * (x + y) = (a * x) + (a * y)
t26_bspace:: for X being set for a, b being Element of Z_2 for x being Element of (bspace X) holds (a + b) * x = (a * x) + (b * x)
t27_bspace:: for X being set for a, b being Element of Z_2 for x being Element of (bspace X) holds (a * b) * x = a * (b * x)
t28_bspace:: for X being set for x being Element of (bspace X) holds (1_ Z_2) * x = x
t29_bspace:: for X being set holds ( bspace X is vector-distributive & bspace X is scalar-distributive & bspace X is scalar-associative & bspace X is scalar-unital )
t3_bspace:: [#] Z_2 = {0,1} by CARD_1:50;
t30_bspace:: for X being non empty set for f being Subset of X st f is Element of singletons X holds f is 1 -element
t31_bspace:: for X being non empty set for x being Element of X holds (<*> (bspace X)) @ x = <*> Z_2
t32_bspace:: for X being set for u, v being Element of (bspace X) for x being Element of X holds (u + v) @ x = (u @ x) + (v @ x)
t33_bspace:: for X being non empty set for s being FinSequence of (bspace X) for f being Element of (bspace X) for x being Element of X holds (s ^ <*f*>) @ x = (s @ x) ^ <*(f @ x)*>
t34_bspace:: for X being non empty set for s being FinSequence of (bspace X) for x being Element of X holds (Sum s) @ x = Sum (s @ x)
t35_bspace:: for X being non empty set for l being Linear_Combination of bspace X for x being Element of (bspace X) st x in Carrier l holds l . x = 1_ Z_2
t36_bspace:: for X being set holds singletons X is linearly-independent
t37_bspace:: for X being set for f being Element of (bspace X) st ex x being set st ( x in X & f = {x} ) holds f in singletons X ;
t38_bspace:: for X being finite set for A being Subset of X ex l being Linear_Combination of singletons X st Sum l = A
t39_bspace:: for X being finite set holds Lin (singletons X) = bspace X
t4_bspace:: for a being Element of Z_2 holds ( a = 0 or a = 1 ) by CARD_1:50, TARSKI:def_2;
t40_bspace:: for X being finite set holds singletons X is Basis of bspace X
t41_bspace:: for X being set holds card (singletons X) = card X
t42_bspace:: for X being set holds card ([#] (bspace X)) = exp (2,(card X)) by CARD_2:31;
t43_bspace:: dim (bspace {}) = 0
t5_bspace:: 0. Z_2 = 0
t6_bspace:: 1. Z_2 = 1 by INT_3:14;
t7_bspace:: (1. Z_2) + (1. Z_2) = 0. Z_2
t8_bspace:: for x being Element of Z_2 holds ( x = 0. Z_2 iff x <> 1. Z_2 ) by Th5, Th6, CARD_1:50, TARSKI:def_2;
t9_bspace:: for X, x being set holds ( X @ x = 1. Z_2 iff x in X ) by Def3;
t1_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a '&' b) 'or' (b '&' c)) 'or' (c '&' a) = ((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)
t10_bvfunc10:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' d) = ((a 'imp' ((b '&' c) '&' d)) '&' (b 'imp' (c '&' d))) '&' (c 'imp' d)
t11_bvfunc10:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds ((a 'imp' c) '&' (b 'imp' d)) '&' (a 'or' b) '<' c 'or' d
t12_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (((a '&' b) 'imp' ('not' c)) '&' a) '&' c '<' 'not' b
t13_bvfunc10:: for Y being non empty set for a1, a2, a3, b1, b2, b3 being Function of Y,BOOLEAN holds ((a1 '&' a2) '&' a3) 'imp' ((b1 'or' b2) 'or' b3) = ((('not' b1) '&' ('not' b2)) '&' a3) 'imp' ((('not' a1) 'or' ('not' a2)) 'or' b3)
t14_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a) = ((a '&' b) '&' c) 'or' ((('not' a) '&' ('not' b)) '&' ('not' c))
t15_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' ((a 'or' b) 'or' c) = (a '&' b) '&' c
t16_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (((a 'or' b) '&' (b 'or' c)) '&' (c 'or' a)) '&' ('not' ((a '&' b) '&' c)) = (((('not' a) '&' b) '&' c) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' b) '&' ('not' c))
t17_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' a 'imp' (b '&' c)
t18_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'or' b) 'imp' c
t19_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' a 'imp' (b 'or' c)
t2_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a '&' ('not' b)) 'or' (b '&' ('not' c))) 'or' (c '&' ('not' a)) = ((b '&' ('not' a)) 'or' (c '&' ('not' b))) 'or' (a '&' ('not' c))
t20_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' a 'imp' (b 'or' ('not' c))
t21_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' b 'imp' (c 'or' a)
t22_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' b 'imp' (c 'or' ('not' a))
t23_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' b) '&' (b 'imp' (c 'or' a))
t24_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' ('not' c))) '&' (b 'imp' c)
t25_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' c)) '&' (b 'imp' (c 'or' a))
t26_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' ('not' c))) '&' (b 'imp' (c 'or' a))
t27_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) '<' (a 'imp' (b 'or' ('not' c))) '&' (b 'imp' (c 'or' ('not' a)))
t3_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'or' ('not' b)) '&' (b 'or' ('not' c))) '&' (c 'or' ('not' a)) = ((b 'or' ('not' a)) '&' (c 'or' ('not' b))) '&' (a 'or' ('not' c))
t4_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st c 'imp' a = I_el Y & c 'imp' b = I_el Y holds c 'imp' (a 'or' b) = I_el Y
t5_bvfunc10:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' c = I_el Y & b 'imp' c = I_el Y holds (a '&' b) 'imp' c = I_el Y
t6_bvfunc10:: for Y being non empty set for a1, a2, b1, b2, c1, c2 being Function of Y,BOOLEAN holds (((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1) '<' (a2 'or' b2) 'or' c2
t7_bvfunc10:: for Y being non empty set for a1, a2, b1, b2 being Function of Y,BOOLEAN holds (((a1 'imp' b1) '&' (a2 'imp' b2)) '&' (a1 'or' a2)) '&' ('not' (b1 '&' b2)) = (((b1 'imp' a1) '&' (b2 'imp' a2)) '&' (b1 'or' b2)) '&' ('not' (a1 '&' a2))
t8_bvfunc10:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds (a 'or' b) '&' (c 'or' d) = (((a '&' c) 'or' (a '&' d)) 'or' (b '&' c)) 'or' (b '&' d)
t9_bvfunc10:: for Y being non empty set for a1, a2, b1, b2, b3 being Function of Y,BOOLEAN holds (a1 '&' a2) 'or' ((b1 '&' b2) '&' b3) = (((((a1 'or' b1) '&' (a1 'or' b2)) '&' (a1 'or' b3)) '&' (a2 'or' b1)) '&' (a2 'or' b2)) '&' (a2 'or' b3)
t1_bvfunc11:: for Y being non empty set for z being Element of Y for PA, PB being a_partition of Y st PA '<' PB holds EqClass (z,PA) c= EqClass (z,PB)
t10_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((All (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
t11_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((All (a,A,G)),B,G) '<' All ((Ex (a,B,G)),A,G)
t12_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((Ex (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
t13_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),B,G)),A,G)
t14_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (All (a,A,G))),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
t15_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((All (a,A,G)),B,G)) = Ex (('not' (All (a,B,G))),A,G)
t16_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (All (a,A,G))),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
t17_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),B,G)),A,G)
t18_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((All (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),A,G)),B,G)
t19_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (All ((Ex (a,A,G)),B,G)) = Ex ((All (('not' a),A,G)),B,G)
t2_bvfunc11:: for Y being non empty set for z being Element of Y for PA, PB being a_partition of Y holds EqClass (z,PA) c= EqClass (z,(PA '\/' PB)) by Th1, PARTIT1:16;
t20_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((All (a,A,G)),B,G)) = All ((Ex (('not' a),A,G)),B,G)
t21_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (All ((All (a,A,G)),B,G)) = Ex ((Ex (('not' a),A,G)),B,G)
t22_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex ((All (a,A,G)),B,G) '<' Ex ((Ex (a,B,G)),A,G)
t23_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((All (a,A,G)),B,G) '<' All ((Ex (a,A,G)),B,G)
t24_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((All (a,A,G)),B,G) '<' Ex ((Ex (a,A,G)),B,G)
t25_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds Ex ((All (a,A,G)),B,G) '<' Ex ((Ex (a,A,G)),B,G)
t26_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (All (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
t27_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
t28_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
t29_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((Ex (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
t3_bvfunc11:: for Y being non empty set for z being Element of Y for PA, PB being a_partition of Y holds EqClass (z,(PA '/\' PB)) c= EqClass (z,PA) by Th1, PARTIT1:15;
t30_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex ((All (('not' a),A,G)),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
t31_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((All (a,B,G)),A,G))
t32_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((All (a,B,G)),A,G)) by PARTIT_2:11, PARTIT_2:17;
t33_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (Ex ((All (a,B,G)),A,G))
t34_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (All ((Ex (a,B,G)),A,G))
t35_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (Ex ((All (a,A,G)),B,G)) '<' 'not' (All ((All (a,B,G)),A,G))
t36_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((Ex (a,A,G)),B,G)) '<' 'not' (All ((All (a,B,G)),A,G))
t37_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' 'not' (All ((All (a,B,G)),A,G))
t38_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (Ex ((All (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G)
t39_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((Ex (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G)
t4_bvfunc11:: for Y being non empty set for z being Element of Y for PA being a_partition of Y holds ( EqClass (z,PA) c= EqClass (z,(%O Y)) & EqClass (z,(%I Y)) c= EqClass (z,PA) ) by Th1, PARTIT1:32;
t40_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (All (a,B,G))),A,G)
t41_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((Ex (a,A,G)),B,G)) '<' All (('not' (All (a,B,G))),A,G)
t42_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' All (('not' (All (a,B,G))),A,G)
t43_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex (('not' (Ex (a,B,G))),A,G)
t44_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (Ex ((Ex (a,A,G)),B,G)) = All (('not' (Ex (a,B,G))),A,G)
t45_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((Ex (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),B,G)),A,G)
t46_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex ((Ex (('not' a),B,G)),A,G)
t47_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (All ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' a),B,G)),A,G)
t48_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' All ((Ex (('not' a),B,G)),A,G)
t49_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds 'not' (Ex ((Ex (a,A,G)),B,G)) '<' Ex ((All (('not' a),B,G)),A,G)
t5_bvfunc11:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent & G = {A,B} & A <> B holds for a, b being set st a in A & b in B holds a meets b
t50_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds 'not' (Ex ((Ex (a,A,G)),B,G)) = All ((All (('not' a),B,G)),A,G)
t51_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
t52_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
t53_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' 'not' (All ((Ex (a,B,G)),A,G))
t54_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (Ex (a,A,G))),B,G) = 'not' (Ex ((Ex (a,B,G)),A,G))
t55_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (All (a,A,G))),B,G) = Ex (('not' (All (a,B,G))),A,G)
t56_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (All (a,A,G))),B,G) '<' Ex (('not' (All (a,B,G))),A,G)
t57_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (Ex (a,A,G))),B,G) '<' Ex (('not' (All (a,B,G))),A,G)
t58_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' Ex (('not' (All (a,B,G))),A,G)
t59_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (Ex (a,A,G))),B,G) '<' All (('not' (All (a,B,G))),A,G)
t6_bvfunc11:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent & G = {A,B} & A <> B holds '/\' G = A '/\' B
t60_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (Ex (a,A,G))),B,G) '<' All (('not' (All (a,B,G))),A,G)
t61_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' Ex (('not' (Ex (a,B,G))),A,G)
t62_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (Ex (a,A,G))),B,G) = All (('not' (Ex (a,B,G))),A,G)
t63_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (Ex (a,A,G))),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
t64_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
t65_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex (('not' (Ex (a,A,G))),B,G) '<' All ((Ex (('not' a),B,G)),A,G)
t66_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (Ex (a,A,G))),B,G) '<' All ((Ex (('not' a),B,G)),A,G)
t67_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All (('not' (Ex (a,A,G))),B,G) '<' Ex ((All (('not' a),B,G)),A,G)
t68_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All (('not' (Ex (a,A,G))),B,G) = All ((All (('not' a),B,G)),A,G)
t69_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
t7_bvfunc11:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G = {A,B} & A <> B holds CompF (A,G) = B
t70_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((All (a,B,G)),A,G))
t71_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((All (('not' a),A,G)),B,G) '<' 'not' (All ((Ex (a,B,G)),A,G))
t72_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((All (('not' a),A,G)),B,G) '<' 'not' (Ex ((Ex (a,B,G)),A,G))
t73_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex ((Ex (('not' a),A,G)),B,G) '<' Ex (('not' (All (a,B,G))),A,G)
t74_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((Ex (('not' a),A,G)),B,G) '<' Ex (('not' (All (a,B,G))),A,G)
t75_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex ((All (('not' a),A,G)),B,G) '<' Ex (('not' (All (a,B,G))),A,G)
t76_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((All (('not' a),A,G)),B,G) '<' Ex (('not' (All (a,B,G))),A,G)
t77_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex ((All (('not' a),A,G)),B,G) '<' All (('not' (All (a,B,G))),A,G)
t78_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((All (('not' a),A,G)),B,G) '<' All (('not' (All (a,B,G))),A,G)
t79_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((All (('not' a),A,G)),B,G) '<' Ex (('not' (Ex (a,B,G))),A,G)
t8_bvfunc11:: for Y being non empty set for a, b being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A being a_partition of Y st a '<' b holds All (a,A,G) '<' Ex (b,A,G)
t80_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((All (('not' a),A,G)),B,G) = All (('not' (Ex (a,B,G))),A,G)
t81_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y holds All ((Ex (('not' a),A,G)),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
t82_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds Ex ((All (('not' a),A,G)),B,G) '<' Ex ((Ex (('not' a),B,G)),A,G)
t9_bvfunc11:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G is independent holds All ((All (a,A,G)),B,G) '<' Ex ((All (a,B,G)),A,G)
t1_bvfunc14:: for Y being non empty set for z being Element of Y for PA, PB being a_partition of Y holds EqClass (z,(PA '/\' PB)) = (EqClass (z,PA)) /\ (EqClass (z,PB))
t10_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> D & B <> D & C <> D holds CompF (D,G) = (A '/\' C) '/\' B
t11_bvfunc14:: for B, C, D, b, c, d being set holds dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {B,C,D} by Lm1;
t12_bvfunc14:: for f being Function for C, D, c, d being set st C <> D holds ((f +* (C .--> c)) +* (D .--> d)) . C = c by Lm2;
t13_bvfunc14:: for B, C, D, b, c, d being set st B <> C & D <> B holds (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b by Lm3;
t14_bvfunc14:: for B, C, D, b, c, d being set for h being Function st h = ((B .--> b) +* (C .--> c)) +* (D .--> d) holds rng h = {(h . B),(h . C),(h . D)} by Lm4;
t15_bvfunc14:: for Y being non empty set for A, B, C, D being a_partition of Y for h being Function for A9, B9, C9, D9 being set st A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 )
t16_bvfunc14:: for A, B, C, D being set for h being Function for A9, B9, C9, D9 being set st h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds dom h = {A,B,C,D}
t17_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for h being Function for A9, B9, C9, D9 being set st G = {A,B,C,D} & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D)}
t18_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D holds EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A)
t19_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & EqClass (z,(C '/\' D)) = EqClass (u,(C '/\' D)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G)))
t2_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G = {A,B} & A <> B holds '/\' G = A '/\' B
t20_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C} & A <> B & B <> C & C <> A & EqClass (z,C) = EqClass (u,C) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G)))
t21_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E holds CompF (A,G) = ((B '/\' C) '/\' D) '/\' E
t22_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & B <> C & B <> D & B <> E holds CompF (B,G) = ((A '/\' C) '/\' D) '/\' E
t23_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> C & B <> C & C <> D & C <> E holds CompF (C,G) = ((A '/\' B) '/\' D) '/\' E
t24_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> D & B <> D & C <> D & D <> E holds CompF (D,G) = ((A '/\' B) '/\' C) '/\' E
t25_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E holds CompF (E,G) = ((A '/\' B) '/\' C) '/\' D
t26_bvfunc14:: for A, B, C, D, E being set for h being Function for A9, B9, C9, D9, E9 being set st A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 )
t27_bvfunc14:: for A, B, C, D, E being set for h being Function for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds dom h = {A,B,C,D,E}
t28_bvfunc14:: for A, B, C, D, E being set for h being Function for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E)}
t29_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E holds EqClass (u,(((B '/\' C) '/\' D) '/\' E)) meets EqClass (z,A)
t3_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for B, C, D being a_partition of Y st G = {B,C,D} & B <> C & C <> D & D <> B holds '/\' G = (B '/\' C) '/\' D
t30_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & EqClass (z,((C '/\' D) '/\' E)) = EqClass (u,((C '/\' D) '/\' E)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G)))
t31_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (A,G) = (((B '/\' C) '/\' D) '/\' E) '/\' F
t32_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F
t33_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (C,G) = (((A '/\' B) '/\' D) '/\' E) '/\' F
t34_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (D,G) = (((A '/\' B) '/\' C) '/\' E) '/\' F
t35_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F
t36_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (F,G) = (((A '/\' B) '/\' C) '/\' D) '/\' E
t37_bvfunc14:: for A, B, C, D, E, F being set for h being Function for A9, B9, C9, D9, E9, F9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 )
t38_bvfunc14:: for A, B, C, D, E, F being set for h being Function for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F}
t39_bvfunc14:: for A, B, C, D, E, F being set for h being Function for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)}
t4_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & C <> A holds CompF (A,G) = B '/\' C
t40_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A)
t41_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G)))
t42_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J
t43_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (B,G) = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J
t44_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (C,G) = ((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J
t45_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J
t46_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (E,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J
t47_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J
t48_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (J,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F
t49_bvfunc14:: for A, B, C, D, E, F, J being set for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 )
t5_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & B <> C holds CompF (B,G) = C '/\' A
t50_bvfunc14:: for A, B, C, D, E, F, J being set for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J}
t51_bvfunc14:: for A, B, C, D, E, F, J being set for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)}
t52_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A)
t53_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) = EqClass (u,((((C '/\' D) '/\' E) '/\' F) '/\' J)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G)))
t54_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (A,G) = (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
t55_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (B,G) = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
t56_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
t57_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (D,G) = (((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M
t58_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M
t59_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (F,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M
t6_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & B <> C & C <> A holds CompF (C,G) = A '/\' B
t60_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (J,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M
t61_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (M,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J
t62_bvfunc14:: for A, B, C, D, E, F, J, M being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 )
t63_bvfunc14:: for A, B, C, D, E, F, J, M being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M}
t64_bvfunc14:: for A, B, C, D, E, F, J, M being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)}
t65_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {}
t66_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G)))
t67_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N
t68_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (B,G) = ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N
t69_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (C,G) = ((((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N
t7_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & A <> C & A <> D holds CompF (A,G) = (B '/\' C) '/\' D
t70_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (D,G) = ((((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N
t71_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (E,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M) '/\' N
t72_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N
t73_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (J,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M) '/\' N
t74_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (M,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' N
t75_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (N,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
t76_bvfunc14:: for A, B, C, D, E, F, J, M, N being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 & h . M = M9 & h . N = N9 )
t77_bvfunc14:: for A, B, C, D, E, F, J, M, N being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M,N}
t78_bvfunc14:: for A, B, C, D, E, F, J, M, N being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)}
t79_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {}
t8_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & B <> C & B <> D holds CompF (B,G) = (A '/\' C) '/\' D
t80_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G)))
t9_bvfunc14:: for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> C & B <> C & C <> D holds CompF (C,G) = (A '/\' B) '/\' D
t1_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'imp' b) = a '&' ('not' b)
t10_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = a 'xor' ('not' b)
t11_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a '&' (b 'xor' c) = (a '&' b) 'xor' (a '&' c)
t12_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = 'not' (a 'xor' b)
t13_bvfunc25:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'xor' a = O_el Y
t14_bvfunc25:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'xor' ('not' a) = I_el Y
t15_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (b 'imp' a) = b 'imp' a
t16_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'or' b) '&' (('not' a) 'or' ('not' b)) = (('not' a) '&' b) 'or' (a '&' ('not' b))
t17_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a '&' b) 'or' (('not' a) '&' ('not' b)) = (('not' a) 'or' b) '&' (a 'or' ('not' b))
t18_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'xor' (b 'xor' c) = (a 'xor' b) 'xor' c
t19_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' (b 'eqv' c) = (a 'eqv' b) 'eqv' c
t2_bvfunc25:: for Y being non empty set for b, a being Function of Y,BOOLEAN holds (('not' b) 'imp' ('not' a)) 'imp' (a 'imp' b) = I_el Y
t20_bvfunc25:: for Y being non empty set for a being Function of Y,BOOLEAN holds ('not' ('not' a)) 'imp' a = I_el Y by BVFUNC_5:7;
t21_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ((a 'imp' b) '&' a) 'imp' b = I_el Y
t22_bvfunc25:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' (('not' a) 'imp' a) = I_el Y
t23_bvfunc25:: for Y being non empty set for a being Function of Y,BOOLEAN holds (('not' a) 'imp' a) 'eqv' a = I_el Y
t24_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' (a 'imp' b) = I_el Y
t25_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'or' (c 'imp' a) = I_el Y
t26_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) 'or' (('not' a) 'imp' b) = I_el Y
t27_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) 'or' (a 'imp' ('not' b)) = I_el Y
t28_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ('not' a) 'imp' (('not' b) 'eqv' (b 'imp' a)) = I_el Y
t29_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (((a 'imp' c) 'imp' b) 'imp' b) = I_el Y
t3_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' b = ('not' b) 'imp' ('not' a)
t30_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' b = a 'eqv' (a '&' b)
t31_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( ( a 'imp' b = I_el Y & b 'imp' a = I_el Y ) iff a = b )
t32_bvfunc25:: for Y being non empty set for a being Function of Y,BOOLEAN holds a = ('not' a) 'imp' a
t33_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' ((a 'imp' b) 'imp' a) = I_el Y
t34_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a = (a 'imp' b) 'imp' a
t35_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a = (b 'imp' a) '&' (('not' b) 'imp' a)
t36_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' b = 'not' (a 'imp' ('not' b))
t37_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' b = ('not' a) 'imp' b
t38_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' b = (a 'imp' b) 'imp' b
t39_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (a 'imp' a) = I_el Y
t4_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = ('not' a) 'eqv' ('not' b)
t40_bvfunc25:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds (a 'imp' (b 'imp' c)) 'imp' ((d 'imp' b) 'imp' (a 'imp' (d 'imp' c))) = I_el Y
t41_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (((a 'imp' b) '&' a) '&' c) 'imp' b = I_el Y
t42_bvfunc25:: for Y being non empty set for b, c, a being Function of Y,BOOLEAN holds (b 'imp' c) 'imp' ((a '&' b) 'imp' c) = I_el Y
t43_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a '&' b) 'imp' c) 'imp' ((a '&' b) 'imp' (c '&' b)) = I_el Y
t44_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((a '&' c) 'imp' (b '&' c)) = I_el Y
t45_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'imp' b) '&' (a '&' c)) 'imp' (b '&' c) = I_el Y
t46_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' (a 'imp' b)) '&' (b 'imp' c) '<' c
t47_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'or' b) '&' (a 'imp' c)) '&' (b 'imp' c) '<' ('not' a) 'imp' (b 'or' c)
t5_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' b = a 'imp' (a '&' b)
t6_bvfunc25:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a 'or' b) 'imp' (a '&' b)
t7_bvfunc25:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'eqv' ('not' a) = O_el Y
t8_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b 'imp' c) = b 'imp' (a 'imp' c)
t9_bvfunc25:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b 'imp' c) = (a 'imp' b) 'imp' (a 'imp' c)
d1_bvfunc26:: for p, q being boolean-valued Function for b3 being Function holds ( b3 = p 'nand' q iff ( dom b3 = (dom p) /\ (dom q) & ( for x being set st x in dom b3 holds b3 . x = (p . x) 'nand' (q . x) ) ) );
d2_bvfunc26:: for p, q being boolean-valued Function for b3 being Function holds ( b3 = p 'nor' q iff ( dom b3 = (dom p) /\ (dom q) & ( for x being set st x in dom b3 holds b3 . x = (p . x) 'nor' (q . x) ) ) );
d3_bvfunc26:: for A being non empty set for p, q, b4 being Function of A,BOOLEAN holds ( b4 = p 'nand' q iff for x being Element of A holds b4 . x = (p . x) 'nand' (q . x) );
d4_bvfunc26:: for A being non empty set for p, q, b4 being Function of A,BOOLEAN holds ( b4 = p 'nor' q iff for x being Element of A holds b4 . x = (p . x) 'nor' (q . x) );
t1_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' b = 'not' (a '&' b)
t10_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nand' (b '&' c) = (a '&' b) 'nand' c
t11_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nand' (b 'or' c) = ('not' (a '&' b)) '&' ('not' (a '&' c))
t12_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nand' (b 'xor' c) = (a '&' b) 'eqv' (a '&' c)
t13_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ( a 'nand' (b 'nand' c) = ('not' a) 'or' (b '&' c) & a 'nand' (b 'nand' c) = a 'imp' (b '&' c) )
t14_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ( a 'nand' (b 'nor' c) = (('not' a) 'or' b) 'or' c & a 'nand' (b 'nor' c) = a 'imp' (b 'or' c) )
t15_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nand' (b 'eqv' c) = a 'imp' (b 'xor' c)
t16_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' (a '&' b) = a 'nand' b
t17_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' (a 'or' b) = ('not' a) '&' ('not' (a '&' b))
t18_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' (a 'eqv' b) = a 'imp' (a 'xor' b)
t19_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( a 'nand' (a 'nand' b) = ('not' a) 'or' b & a 'nand' (a 'nand' b) = a 'imp' b )
t2_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' b = 'not' (a 'or' b)
t20_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' (a 'nor' b) = I_el Y
t21_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' (a 'eqv' b) = ('not' a) 'or' ('not' b)
t22_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' b = (a 'nand' b) 'nand' (a 'nand' b)
t23_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'nand' b) 'nand' (a 'nand' c) = a '&' (b 'or' c)
t24_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nand' (b 'imp' c) = (('not' a) 'or' b) '&' ('not' (a '&' c))
t25_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' (a 'imp' b) = 'not' (a '&' b)
t26_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds (I_el Y) 'nor' a = O_el Y
t27_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds (O_el Y) 'nor' a = 'not' a
t28_bvfunc26:: for Y being non empty set holds ( (O_el Y) 'nor' (O_el Y) = I_el Y & (O_el Y) 'nor' (I_el Y) = O_el Y & (I_el Y) 'nor' (I_el Y) = O_el Y )
t29_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( a 'nor' a = 'not' a & 'not' (a 'nor' a) = a )
t3_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds (I_el Y) 'nand' a = 'not' a
t30_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'nor' b) = a 'or' b
t31_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( a 'nor' ('not' a) = O_el Y & 'not' (a 'nor' ('not' a)) = I_el Y )
t32_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ('not' a) '&' (a 'xor' b) = ('not' a) '&' b
t33_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nor' (b '&' c) = ('not' (a 'or' b)) 'or' ('not' (a 'or' c))
t34_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nor' (b 'or' c) = 'not' ((a 'or' b) 'or' c)
t35_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nor' (b 'eqv' c) = ('not' a) '&' (b 'xor' c)
t36_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nor' (b 'imp' c) = (('not' a) '&' b) '&' ('not' c)
t37_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nor' (b 'nand' c) = (('not' a) '&' b) '&' c
t38_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nor' (b 'nor' c) = ('not' a) '&' (b 'or' c)
t39_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' (a '&' b) = 'not' (a '&' (a 'or' b))
t4_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds (O_el Y) 'nand' a = I_el Y
t40_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' (a 'or' b) = 'not' (a 'or' b)
t41_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' (a 'eqv' b) = ('not' a) '&' b
t42_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' (a 'imp' b) = O_el Y
t43_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' (a 'nand' b) = O_el Y
t44_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' (a 'nor' b) = ('not' a) '&' b
t45_bvfunc26:: for Y being non empty set holds (O_el Y) 'eqv' (O_el Y) = I_el Y
t46_bvfunc26:: for Y being non empty set holds (O_el Y) 'eqv' (I_el Y) = O_el Y
t47_bvfunc26:: for Y being non empty set holds (I_el Y) 'eqv' (I_el Y) = I_el Y
t48_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( a 'eqv' a = I_el Y & 'not' (a 'eqv' a) = O_el Y )
t49_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' (a 'or' b) = a 'or' ('not' b)
t5_bvfunc26:: for Y being non empty set holds ( (O_el Y) 'nand' (O_el Y) = I_el Y & (O_el Y) 'nand' (I_el Y) = I_el Y & (I_el Y) 'nand' (I_el Y) = O_el Y )
t50_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a '&' (b 'nand' c) = (a '&' ('not' b)) 'or' (a '&' ('not' c))
t51_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'or' (b 'nand' c) = (a 'or' ('not' b)) 'or' ('not' c)
t52_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'xor' (b 'nand' c) = (('not' a) '&' ('not' (b '&' c))) 'or' ((a '&' b) '&' c)
t53_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' (b 'nand' c) = (a '&' ('not' (b '&' c))) 'or' ((('not' a) '&' b) '&' c)
t54_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b 'nand' c) = 'not' ((a '&' b) '&' c)
t55_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nor' (b 'nand' c) = 'not' ((a 'or' ('not' b)) 'or' ('not' c))
t56_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' (a 'nand' b) = a '&' ('not' b)
t57_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' (a 'nand' b) = I_el Y
t58_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' (a 'nand' b) = ('not' a) 'or' b
t59_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' (a 'nand' b) = a '&' ('not' b)
t6_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( a 'nand' a = 'not' a & 'not' (a 'nand' a) = a )
t60_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' (a 'nand' b) = 'not' (a '&' b)
t61_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nor' (a 'nand' b) = O_el Y
t62_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a '&' (b 'nor' c) = (a '&' ('not' b)) '&' ('not' c)
t63_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'or' (b 'nor' c) = (a 'or' ('not' b)) '&' (a 'or' ('not' c))
t64_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'xor' (b 'nor' c) = (a 'or' ('not' (b 'or' c))) '&' ((('not' a) 'or' b) 'or' c)
t65_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' (b 'nor' c) = ((a 'or' b) 'or' c) '&' (('not' a) 'or' ('not' (b 'or' c)))
t66_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b 'nor' c) = 'not' (a '&' (b 'or' c))
t67_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nand' (b 'nor' c) = (('not' a) 'or' b) 'or' c
t68_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' (a 'nor' b) = O_el Y
t69_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' (a 'nor' b) = a 'or' ('not' b)
t7_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'nand' b) = a '&' b
t70_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' (a 'nor' b) = a 'or' ('not' b)
t71_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' (a 'nor' b) = ('not' a) '&' b
t72_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' (a 'nor' b) = 'not' (a 'or' (a '&' b))
t73_bvfunc26:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'nand' (a 'nor' b) = I_el Y
t8_bvfunc26:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( a 'nand' ('not' a) = I_el Y & 'not' (a 'nand' ('not' a)) = O_el Y )
t9_bvfunc26:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'nand' (b '&' c) = 'not' ((a '&' b) '&' c)
d1_bvfunc_1:: for k, l being boolean set holds ( k <= l iff k => l = TRUE );
d10_bvfunc_1:: for Y being non empty set for b2 being Function of Y,BOOLEAN holds ( b2 = O_el Y iff for x being Element of Y holds b2 . x = FALSE );
d11_bvfunc_1:: for Y being non empty set for b2 being Function of Y,BOOLEAN holds ( b2 = I_el Y iff for x being Element of Y holds b2 . x = TRUE );
d12_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( a '<' b iff for x being Element of Y st a . x = TRUE holds b . x = TRUE );
d13_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( ( ( for x being Element of Y holds a . x = TRUE ) implies B_INF a = I_el Y ) & ( not for x being Element of Y holds a . x = TRUE implies B_INF a = O_el Y ) );
d14_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( ( ( for x being Element of Y holds a . x = FALSE ) implies B_SUP a = O_el Y ) & ( not for x being Element of Y holds a . x = FALSE implies B_SUP a = I_el Y ) );
d15_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y holds ( a is_dependent_of PA iff for F being set st F in PA holds for x1, x2 being set st x1 in F & x2 in F holds a . x1 = a . x2 );
d16_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y for b4 being Function of Y,BOOLEAN holds ( b4 = B_INF (a,PA) iff for y being Element of Y holds ( ( ( for x being Element of Y st x in EqClass (y,PA) holds a . x = TRUE ) implies b4 . y = TRUE ) & ( ex x being Element of Y st ( x in EqClass (y,PA) & not a . x = TRUE ) implies b4 . y = FALSE ) ) );
d17_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y for b4 being Function of Y,BOOLEAN holds ( b4 = B_SUP (a,PA) iff for y being Element of Y holds ( ( ex x being Element of Y st ( x in EqClass (y,PA) & a . x = TRUE ) implies b4 . y = TRUE ) & ( ( for x being Element of Y holds ( not x in EqClass (y,PA) or not a . x = TRUE ) ) implies b4 . y = FALSE ) ) );
d18_bvfunc_1:: for Y being non empty set for f being Function of Y,BOOLEAN holds GPart f = { { x where x is Element of Y : f . x = TRUE } , { x9 where x9 is Element of Y : f . x9 = FALSE } } \ {{}};
d19_bvfunc_1:: for x, y being boolean set holds x <=> y = 'not' (x 'xor' y);
d2_bvfunc_1:: for p, q, b3 being boolean-valued Function holds ( b3 = p 'or' q iff ( dom b3 = (dom p) /\ (dom q) & ( for x being set st x in dom b3 holds b3 . x = (p . x) 'or' (q . x) ) ) );
d3_bvfunc_1:: for p, q being boolean-valued Function for b3 being Function holds ( b3 = p 'xor' q iff ( dom b3 = (dom p) /\ (dom q) & ( for x being set st x in dom b3 holds b3 . x = (p . x) 'xor' (q . x) ) ) );
d4_bvfunc_1:: for A being non empty set for p, q, b4 being Function of A,BOOLEAN holds ( b4 = p 'or' q iff for x being Element of A holds b4 . x = (p . x) 'or' (q . x) );
d5_bvfunc_1:: for A being non empty set for p, q, b4 being Function of A,BOOLEAN holds ( b4 = p 'xor' q iff for x being Element of A holds b4 . x = (p . x) 'xor' (q . x) );
d6_bvfunc_1:: for p, q being boolean-valued Function for b3 being Function holds ( b3 = p 'imp' q iff ( dom b3 = (dom p) /\ (dom q) & ( for x being set st x in dom b3 holds b3 . x = (p . x) => (q . x) ) ) );
d7_bvfunc_1:: for p, q being boolean-valued Function for b3 being Function holds ( b3 = p 'eqv' q iff ( dom b3 = (dom p) /\ (dom q) & ( for x being set st x in dom b3 holds b3 . x = (p . x) <=> (q . x) ) ) );
d8_bvfunc_1:: for A being non empty set for p, q, b4 being Function of A,BOOLEAN holds ( b4 = p 'imp' q iff for x being Element of A holds b4 . x = ('not' (p . x)) 'or' (q . x) );
d9_bvfunc_1:: for A being non empty set for p, q, b4 being Function of A,BOOLEAN holds ( b4 = p 'eqv' q iff for x being Element of A holds b4 . x = 'not' ((p . x) 'xor' (q . x)) );
s1_bvfunc_1:: scheme BVFUniq1{ F1() -> non empty set , F2( set ) -> set } : for f1, f2 being Function of F1(),BOOLEAN st ( for x being Element of F1() holds f1 . x = F2(x) ) & ( for x being Element of F1() holds f2 . x = F2(x) ) holds f1 = f2
t1_bvfunc_1:: canceled;
t10_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'or' (I_el Y) = I_el Y
t11_bvfunc_1:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'or' c = (a 'or' c) '&' (b 'or' c)
t12_bvfunc_1:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) '&' c = (a '&' c) 'or' (b '&' c)
t13_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'or' b) = ('not' a) '&' ('not' b)
t14_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a '&' b) = ('not' a) 'or' ('not' b)
t15_bvfunc_1:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ( ( a '<' b & b '<' a implies a = b ) & ( a '<' b & b '<' c implies a '<' c ) )
t16_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( a 'imp' b = I_el Y iff a '<' b )
t17_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( a 'eqv' b = I_el Y iff a = b )
t18_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( O_el Y '<' a & a '<' I_el Y )
t19_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds ( 'not' (B_INF a) = B_SUP ('not' a) & 'not' (B_SUP a) = B_INF ('not' a) )
t2_bvfunc_1:: for Y being non empty set holds ( 'not' (I_el Y) = O_el Y & 'not' (O_el Y) = I_el Y )
t20_bvfunc_1:: for Y being non empty set holds ( B_INF (O_el Y) = O_el Y & B_INF (I_el Y) = I_el Y & B_SUP (O_el Y) = O_el Y & B_SUP (I_el Y) = I_el Y )
t21_bvfunc_1:: for Y being non empty set for a being constant Function of Y,BOOLEAN holds ( a = O_el Y or a = I_el Y )
t22_bvfunc_1:: for Y being non empty set for d being constant Function of Y,BOOLEAN holds ( B_INF d = d & B_SUP d = d )
t23_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( B_INF (a '&' b) = (B_INF a) '&' (B_INF b) & B_SUP (a 'or' b) = (B_SUP a) 'or' (B_SUP b) )
t24_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for d being constant Function of Y,BOOLEAN holds ( B_INF (d 'imp' a) = d 'imp' (B_INF a) & B_INF (a 'imp' d) = (B_SUP a) 'imp' d )
t25_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for d being constant Function of Y,BOOLEAN holds ( B_INF (d 'or' a) = d 'or' (B_INF a) & B_SUP (d '&' a) = d '&' (B_SUP a) & B_SUP (a '&' d) = (B_SUP a) '&' d )
t26_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for x being Element of Y holds (B_INF a) . x <= a . x
t27_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for x being Element of Y holds a . x <= (B_SUP a) . x
t28_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds a is_dependent_of %I Y
t29_bvfunc_1:: for Y being non empty set for a being constant Function of Y,BOOLEAN holds a is_dependent_of %O Y
t3_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' a = a ;
t30_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y holds B_INF (a,PA) is_dependent_of PA
t31_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y holds B_SUP (a,PA) is_dependent_of PA
t32_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y holds B_INF (a,PA) '<' a
t33_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y holds a '<' B_SUP (a,PA)
t34_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y holds 'not' (B_INF (a,PA)) = B_SUP (('not' a),PA)
t35_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds B_INF (a,(%O Y)) = B_INF a
t36_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds B_SUP (a,(%O Y)) = B_SUP a
t37_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds B_INF (a,(%I Y)) = a
t38_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds B_SUP (a,(%I Y)) = a
t39_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds B_INF ((a '&' b),PA) = (B_INF (a,PA)) '&' (B_INF (b,PA))
t4_bvfunc_1:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' b) '&' c = a '&' (b '&' c)
t40_bvfunc_1:: for Y being non empty set for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds B_SUP ((a 'or' b),PA) = (B_SUP (a,PA)) 'or' (B_SUP (b,PA))
t41_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds a is_dependent_of GPart a
t42_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN for PA being a_partition of Y st a is_dependent_of PA holds PA is_finer_than GPart a
t43_bvfunc_1:: for x being boolean set holds TRUE 'nand' x = 'not' x ;
t44_bvfunc_1:: for x being boolean set holds FALSE 'nand' x = TRUE ;
t45_bvfunc_1:: for x being boolean set holds ( x 'nand' x = 'not' x & 'not' (x 'nand' x) = x ) ;
t46_bvfunc_1:: for x, y being boolean set holds 'not' (x 'nand' y) = x '&' y ;
t47_bvfunc_1:: for x being boolean set holds ( x 'nand' ('not' x) = TRUE & 'not' (x 'nand' ('not' x)) = FALSE ) by XBOOLEAN:135, XBOOLEAN:138;
t48_bvfunc_1:: for x, y, z being boolean set holds x 'nand' (y '&' z) = 'not' ((x '&' y) '&' z) ;
t49_bvfunc_1:: for x, y, z being boolean set holds x 'nand' (y '&' z) = (x '&' y) 'nand' z ;
t5_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds a '&' (O_el Y) = O_el Y
t50_bvfunc_1:: for x being boolean set holds TRUE 'nor' x = FALSE ;
t51_bvfunc_1:: for x being boolean set holds FALSE 'nor' x = 'not' x ;
t52_bvfunc_1:: for x being boolean set holds ( x 'nor' x = 'not' x & 'not' (x 'nor' x) = x ) ;
t53_bvfunc_1:: for x, y being boolean set holds 'not' (x 'nor' y) = x 'or' y ;
t54_bvfunc_1:: for x being boolean set holds ( x 'nor' ('not' x) = FALSE & 'not' (x 'nor' ('not' x)) = TRUE ) by XBOOLEAN:134, XBOOLEAN:138;
t55_bvfunc_1:: for x, y, z being boolean set holds x 'nor' (y 'or' z) = 'not' ((x 'or' y) 'or' z) ;
t56_bvfunc_1:: for x being boolean set holds TRUE <=> x = x ;
t57_bvfunc_1:: for x being boolean set holds FALSE <=> x = 'not' x ;
t58_bvfunc_1:: for x being boolean set holds ( x <=> x = TRUE & 'not' (x <=> x) = FALSE ) by XBOOLEAN:125, XBOOLEAN:143;
t59_bvfunc_1:: for x, y being boolean set holds 'not' (x <=> y) = x 'xor' y ;
t6_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds a '&' (I_el Y) = a
t60_bvfunc_1:: for x being boolean set holds ( x <=> ('not' x) = FALSE & 'not' (x <=> ('not' x)) = TRUE ) by XBOOLEAN:129, XBOOLEAN:142;
t61_bvfunc_1:: for x, y, z being boolean set holds ( x <= y => z iff x '&' y <= z )
t62_bvfunc_1:: for x, y being boolean set holds x <=> y = (x => y) '&' (y => x) ;
t63_bvfunc_1:: for x, y being boolean set holds x => y = ('not' y) => ('not' x) ;
t64_bvfunc_1:: for x, y being boolean set holds x <=> y = ('not' x) <=> ('not' y) ;
t65_bvfunc_1:: for x, y being boolean set st x = TRUE & x => y = TRUE holds y = TRUE ;
t66_bvfunc_1:: for y, x being boolean set st y = TRUE holds x => y = TRUE ;
t67_bvfunc_1:: for x, y being boolean set st 'not' x = TRUE holds x => y = TRUE ;
t68_bvfunc_1:: for z, y, x being boolean set st z => (y => x) = TRUE holds y => (z => x) = TRUE ;
t69_bvfunc_1:: for z, y, x being boolean set st z => (y => x) = TRUE & y = TRUE holds z => x = TRUE ;
t7_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'or' a = a ;
t70_bvfunc_1:: for z, y, x being boolean set st z => (y => x) = TRUE & y = TRUE & z = TRUE holds x = TRUE ;
t8_bvfunc_1:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) 'or' c = a 'or' (b 'or' c)
t9_bvfunc_1:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'or' (O_el Y) = a
d1_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b3 being a_partition of Y holds ( b3 = '/\' G iff for x being set holds ( x in b3 iff ex h being Function ex F being Subset-Family of Y st ( dom h = G & rng h = F & ( for d being set st d in G holds h . d in d ) & x = Intersect F & x <> {} ) ) );
d10_bvfunc_2:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds Ex (a,PA,G) = B_SUP (a,(CompF (PA,G)));
d2_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b being set holds ( b is_upper_min_depend_of G iff ( ( for d being a_partition of Y st d in G holds b is_a_dependent_set_of d ) & ( for e being set st e c= b & ( for d being a_partition of Y st d in G holds e is_a_dependent_set_of d ) holds e = b ) ) );
d3_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b3 being a_partition of Y holds ( ( G <> {} implies ( b3 = '\/' G iff for x being set holds ( x in b3 iff x is_upper_min_depend_of G ) ) ) & ( not G <> {} implies ( b3 = '\/' G iff b3 = %I Y ) ) );
d4_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) holds ( G is generating iff '/\' G = %I Y );
d5_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) holds ( G is independent iff for h being Function for F being Subset-Family of Y st dom h = G & rng h = F & ( for d being set st d in G holds h . d in d ) holds Intersect F <> {} );
d6_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) holds ( G is_a_coordinate iff ( G is independent & G is generating ) );
d7_bvfunc_2:: for Y being non empty set for PA being a_partition of Y for G being Subset of (PARTITIONS Y) holds CompF (PA,G) = '/\' (G \ {PA});
d8_bvfunc_2:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds ( a is_independent_of PA,G iff a is_dependent_of CompF (PA,G) );
d9_bvfunc_2:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds All (a,PA,G) = B_INF (a,(CompF (PA,G)));
t1_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for y being Element of Y ex X being Subset of Y st ( y in X & ex h being Function ex F being Subset-Family of Y st ( dom h = G & rng h = F & ( for d being set st d in G holds h . d in d ) & X = Intersect F & X <> {} ) )
t10_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds Ex ((O_el Y),PA,G) = O_el Y
t11_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds All (a,PA,G) '<' a
t12_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds a '<' Ex (a,PA,G)
t13_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (All (a,PA,G)) 'or' (All (b,PA,G)) '<' All ((a 'or' b),PA,G)
t14_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (All (b,PA,G))
t15_bvfunc_2:: for Y being non empty set for a, b being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds Ex ((a '&' b),PA,G) '<' (Ex (a,PA,G)) '&' (Ex (b,PA,G))
t16_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) 'xor' (Ex (b,PA,G)) '<' Ex ((a 'xor' b),PA,G)
t17_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' Ex ((a 'imp' b),PA,G)
t18_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds 'not' (All (a,PA,G)) = Ex (('not' a),PA,G)
t19_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds 'not' (Ex (a,PA,G)) = All (('not' a),PA,G)
t2_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for y being Element of Y st G <> {} holds ex X being Subset of Y st ( y in X & X is_upper_min_depend_of G )
t20_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((u 'imp' a),PA,G) = u 'imp' (All (a,PA,G))
t21_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((a 'imp' u),PA,G) = (Ex (a,PA,G)) 'imp' u
t22_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((u 'or' a),PA,G) = u 'or' (All (a,PA,G))
t23_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((a 'or' u),PA,G) = (All (a,PA,G)) 'or' u by Th22;
t24_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((a 'or' u),PA,G) '<' (Ex (a,PA,G)) 'or' u
t25_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((u '&' a),PA,G) = u '&' (All (a,PA,G))
t26_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((a '&' u),PA,G) = (All (a,PA,G)) '&' u by Th25;
t27_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, u being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u
t28_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((u 'xor' a),PA,G) '<' u 'xor' (All (a,PA,G))
t29_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((a 'xor' u),PA,G) '<' (All (a,PA,G)) 'xor' u by Th28;
t3_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for PA being a_partition of Y st PA in G holds PA '>' '/\' G
t30_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((u 'eqv' a),PA,G) '<' u 'eqv' (All (a,PA,G))
t31_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds All ((a 'eqv' u),PA,G) '<' (All (a,PA,G)) 'eqv' u by Th30;
t32_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds Ex ((u 'or' a),PA,G) = u 'or' (Ex (a,PA,G))
t33_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds Ex ((a 'or' u),PA,G) = (Ex (a,PA,G)) 'or' u by Th32;
t34_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds Ex ((u '&' a),PA,G) = u '&' (Ex (a,PA,G))
t35_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds Ex ((a '&' u),PA,G) = (Ex (a,PA,G)) '&' u by Th34;
t36_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y holds u 'imp' (Ex (a,PA,G)) '<' Ex ((u 'imp' a),PA,G)
t37_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, u being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' u '<' Ex ((a 'imp' u),PA,G)
t38_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds u 'xor' (Ex (a,PA,G)) '<' Ex ((u 'xor' a),PA,G)
t39_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds (Ex (a,PA,G)) 'xor' u '<' Ex ((a 'xor' u),PA,G) by Th38;
t4_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for PA being a_partition of Y st PA in G holds PA '<' '\/' G
t5_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds All (a,PA,G) is_dependent_of CompF (PA,G)
t6_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds Ex (a,PA,G) is_dependent_of CompF (PA,G)
t7_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((I_el Y),PA,G) = I_el Y
t8_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds Ex ((I_el Y),PA,G) = I_el Y
t9_bvfunc_2:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((O_el Y),PA,G) = O_el Y
t1_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds a 'imp' b '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
t10_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (All (b,PA,G))
t11_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'or' b),PA,G) '<' (Ex (a,PA,G)) 'or' (Ex (b,PA,G))
t12_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All (b,PA,G)) '<' Ex ((a '&' b),PA,G)
t13_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (All (a,PA,G)) '&' (Ex (b,PA,G)) '<' Ex ((a '&' b),PA,G)
t14_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
t15_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'imp' b),PA,G) '<' (Ex (a,PA,G)) 'imp' (Ex (b,PA,G))
t16_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (All (b,PA,G)) '<' All ((a 'imp' b),PA,G)
t17_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds a 'imp' b '<' a 'imp' (Ex (b,PA,G))
t18_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds a 'imp' b '<' (All (a,PA,G)) 'imp' b
t19_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds Ex ((a 'imp' b),PA,G) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
t2_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (All (a,PA,G)) '&' (All (b,PA,G)) '<' a '&' b
t20_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All (a,PA,G) '<' (Ex (b,PA,G)) 'imp' (Ex ((a '&' b),PA,G))
t21_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds Ex ((u 'imp' a),PA,G) '<' u 'imp' (Ex (a,PA,G))
t22_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for u, a being Function of Y,BOOLEAN for PA being a_partition of Y st u is_independent_of PA,G holds Ex ((a 'imp' u),PA,G) '<' (All (a,PA,G)) 'imp' u
t23_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (All (a,PA,G)) 'imp' (Ex (b,PA,G)) = Ex ((a 'imp' b),PA,G)
t24_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (All (a,PA,G)) 'imp' (All (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
t25_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) 'imp' (Ex (b,PA,G)) '<' (All (a,PA,G)) 'imp' (Ex (b,PA,G))
t26_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'imp' b),PA,G) = All ((('not' a) 'or' b),PA,G)
t27_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'imp' b),PA,G) = 'not' (Ex ((a '&' ('not' b)),PA,G))
t28_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds Ex (a,PA,G) '<' 'not' ((All ((a 'imp' b),PA,G)) '&' (All ((a 'imp' ('not' b)),PA,G)))
t29_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds Ex (a,PA,G) '<' 'not' (('not' (Ex ((a '&' b),PA,G))) '&' ('not' (Ex ((a '&' ('not' b)),PA,G))))
t3_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds a '&' b '<' (Ex (a,PA,G)) '&' (Ex (b,PA,G))
t30_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) '&' (All ((a 'imp' b),PA,G)) '<' Ex ((a '&' b),PA,G)
t31_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (Ex (a,PA,G)) '&' ('not' (Ex ((a '&' b),PA,G))) '<' 'not' (All ((a 'imp' b),PA,G))
t32_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, c, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (All ((a 'imp' c),PA,G)) '&' (All ((c 'imp' b),PA,G)) '<' All ((a 'imp' b),PA,G)
t33_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for c, b, a being Function of Y,BOOLEAN for PA being a_partition of Y holds (All ((c 'imp' b),PA,G)) '&' (Ex ((a '&' c),PA,G)) '<' Ex ((a '&' b),PA,G)
t34_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b, c, a being Function of Y,BOOLEAN for PA being a_partition of Y holds (All ((b 'imp' ('not' c)),PA,G)) '&' (All ((a 'imp' c),PA,G)) '<' All ((a 'imp' ('not' b)),PA,G)
t35_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b, c, a being Function of Y,BOOLEAN for PA being a_partition of Y holds (All ((b 'imp' c),PA,G)) '&' (All ((a 'imp' ('not' c)),PA,G)) '<' All ((a 'imp' ('not' b)),PA,G)
t36_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b, c, a being Function of Y,BOOLEAN for PA being a_partition of Y holds (All ((b 'imp' ('not' c)),PA,G)) '&' (Ex ((a '&' c),PA,G)) '<' Ex ((a '&' ('not' b)),PA,G)
t37_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b, c, a being Function of Y,BOOLEAN for PA being a_partition of Y holds (All ((b 'imp' c),PA,G)) '&' (Ex ((a '&' ('not' c)),PA,G)) '<' Ex ((a '&' ('not' b)),PA,G)
t38_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for c, b, a being Function of Y,BOOLEAN for PA being a_partition of Y holds ((Ex (c,PA,G)) '&' (All ((c 'imp' b),PA,G))) '&' (All ((c 'imp' a),PA,G)) '<' Ex ((a '&' b),PA,G)
t39_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b, c, a being Function of Y,BOOLEAN for PA being a_partition of Y holds (All ((b 'imp' c),PA,G)) '&' (All ((c 'imp' ('not' a)),PA,G)) '<' All ((a 'imp' ('not' b)),PA,G)
t4_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds 'not' ((All (a,PA,G)) '&' (All (b,PA,G))) = (Ex (('not' a),PA,G)) 'or' (Ex (('not' b),PA,G))
t40_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for b, c, a being Function of Y,BOOLEAN for PA being a_partition of Y holds ((Ex (b,PA,G)) '&' (All ((b 'imp' c),PA,G))) '&' (All ((c 'imp' a),PA,G)) '<' Ex ((a '&' b),PA,G)
t41_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for c, b, a being Function of Y,BOOLEAN for PA being a_partition of Y holds ((Ex (c,PA,G)) '&' (All ((b 'imp' ('not' c)),PA,G))) '&' (All ((c 'imp' a),PA,G)) '<' Ex ((a '&' ('not' b)),PA,G)
t5_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds 'not' ((Ex (a,PA,G)) '&' (Ex (b,PA,G))) = (All (('not' a),PA,G)) 'or' (All (('not' b),PA,G))
t6_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds (All (a,PA,G)) 'or' (All (b,PA,G)) '<' a 'or' b
t7_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds a 'or' b '<' (Ex (a,PA,G)) 'or' (Ex (b,PA,G))
t8_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds a 'xor' b '<' ('not' ((Ex (('not' a),PA,G)) 'xor' (Ex (b,PA,G)))) 'or' ('not' ((Ex (a,PA,G)) 'xor' (Ex (('not' b),PA,G))))
t9_bvfunc_3:: for Y being non empty set for G being Subset of (PARTITIONS Y) for a, b being Function of Y,BOOLEAN for PA being a_partition of Y holds All ((a 'or' b),PA,G) '<' (All (a,PA,G)) 'or' (Ex (b,PA,G))
t1_bvfunc_4:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a '<' b 'imp' c holds a '&' b '<' c
t10_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( a 'eqv' b = I_el Y iff ( a 'imp' b = I_el Y & b 'imp' a = I_el Y ) )
t11_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN st a 'eqv' b = I_el Y holds ('not' a) 'eqv' ('not' b) = I_el Y
t12_bvfunc_4:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a 'eqv' b = I_el Y & c 'eqv' d = I_el Y holds (a '&' c) 'eqv' (b '&' d) = I_el Y
t13_bvfunc_4:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a 'eqv' b = I_el Y & c 'eqv' d = I_el Y holds (a 'imp' c) 'eqv' (b 'imp' d) = I_el Y
t14_bvfunc_4:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a 'eqv' b = I_el Y & c 'eqv' d = I_el Y holds (a 'or' c) 'eqv' (b 'or' d) = I_el Y
t15_bvfunc_4:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a 'eqv' b = I_el Y & c 'eqv' d = I_el Y holds (a 'eqv' c) 'eqv' (b 'eqv' d) = I_el Y
t16_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds All ((a 'eqv' b),PA,G) = (All ((a 'imp' b),PA,G)) '&' (All ((b 'imp' a),PA,G))
t17_bvfunc_4:: for Y being non empty set for a being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA, PB being a_partition of Y holds All (a,PA,G) '<' Ex (a,PB,G)
t18_bvfunc_4:: for Y being non empty set for a, u being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y st a 'imp' u = I_el Y holds (All (a,PA,G)) 'imp' u = I_el Y
t19_bvfunc_4:: for Y being non empty set for u being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y st u is_independent_of PA,G holds Ex (u,PA,G) '<' u
t2_bvfunc_4:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a '&' b '<' c holds a '<' b 'imp' c
t20_bvfunc_4:: for Y being non empty set for u being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y st u is_independent_of PA,G holds u '<' All (u,PA,G)
t21_bvfunc_4:: for Y being non empty set for u being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA, PB being a_partition of Y st u is_independent_of PB,G holds All (u,PA,G) '<' All (u,PB,G)
t22_bvfunc_4:: for Y being non empty set for u being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA, PB being a_partition of Y st u is_independent_of PA,G holds Ex (u,PA,G) '<' Ex (u,PB,G)
t23_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds All ((a 'eqv' b),PA,G) '<' (All (a,PA,G)) 'eqv' (All (b,PA,G))
t24_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds All ((a '&' b),PA,G) '<' a '&' (All (b,PA,G))
t25_bvfunc_4:: for Y being non empty set for a, u being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y holds (All (a,PA,G)) 'imp' u '<' Ex ((a 'imp' u),PA,G)
t26_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y st a 'eqv' b = I_el Y holds (All (a,PA,G)) 'eqv' (All (b,PA,G)) = I_el Y
t27_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN for G being Subset of (PARTITIONS Y) for PA being a_partition of Y st a 'eqv' b = I_el Y holds (Ex (a,PA,G)) 'eqv' (Ex (b,PA,G)) = I_el Y
t3_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' (a '&' b) = a
t4_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' (a 'or' b) = a
t5_bvfunc_4:: for Y being non empty set for a being Function of Y,BOOLEAN holds a '&' ('not' a) = O_el Y
t6_bvfunc_4:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'or' ('not' a) = I_el Y
t7_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a 'imp' b) '&' (b 'imp' a)
t8_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' b = ('not' a) 'or' b
t9_bvfunc_4:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' b = (('not' a) '&' b) 'or' (a '&' ('not' b))
t1_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( ( a = I_el Y & b = I_el Y ) iff a '&' b = I_el Y )
t10_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN st a 'imp' b = I_el Y & a 'imp' ('not' b) = I_el Y holds 'not' a = I_el Y
t11_bvfunc_5:: for Y being non empty set for a being Function of Y,BOOLEAN holds (('not' a) 'imp' a) 'imp' a = I_el Y
t12_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (('not' (b '&' c)) 'imp' ('not' (a '&' c))) = I_el Y
t13_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((b 'imp' c) 'imp' (a 'imp' c)) = I_el Y
t14_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' b = I_el Y holds (b 'imp' c) 'imp' (a 'imp' c) = I_el Y
t15_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds b 'imp' (a 'imp' b) = I_el Y
t16_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'imp' b) 'imp' c) 'imp' (b 'imp' c) = I_el Y
t17_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds b 'imp' ((b 'imp' a) 'imp' a) = I_el Y
t18_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (c 'imp' (b 'imp' a)) 'imp' (b 'imp' (c 'imp' a)) = I_el Y
t19_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (b 'imp' c) 'imp' ((a 'imp' b) 'imp' (a 'imp' c)) = I_el Y
t2_bvfunc_5:: for Y being non empty set for b being Function of Y,BOOLEAN st (I_el Y) 'imp' b = I_el Y holds b = I_el Y
t20_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (b 'imp' (b 'imp' c)) 'imp' (b 'imp' c) = I_el Y
t21_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' (b 'imp' c)) 'imp' ((a 'imp' b) 'imp' (a 'imp' c)) = I_el Y
t22_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN st a = I_el Y holds (a 'imp' b) 'imp' b = I_el Y
t23_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st c 'imp' (b 'imp' a) = I_el Y holds b 'imp' (c 'imp' a) = I_el Y
t24_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st c 'imp' (b 'imp' a) = I_el Y & b = I_el Y holds c 'imp' a = I_el Y
t25_bvfunc_5:: for Y being non empty set for a being Function of Y,BOOLEAN st (I_el Y) 'imp' ((I_el Y) 'imp' a) = I_el Y holds a = I_el Y
t26_bvfunc_5:: for Y being non empty set for b, c being Function of Y,BOOLEAN st b 'imp' (b 'imp' c) = I_el Y holds b 'imp' c = I_el Y
t27_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' (b 'imp' c) = I_el Y holds (a 'imp' b) 'imp' (a 'imp' c) = I_el Y
t28_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' (b 'imp' c) = I_el Y & a 'imp' b = I_el Y holds a 'imp' c = I_el Y
t29_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' (b 'imp' c) = I_el Y & a 'imp' b = I_el Y & a = I_el Y holds c = I_el Y
t3_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN st a = I_el Y holds a 'or' b = I_el Y
t30_bvfunc_5:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a 'imp' (b 'imp' c) = I_el Y & a 'imp' (c 'imp' d) = I_el Y holds a 'imp' (b 'imp' d) = I_el Y
t31_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (('not' a) 'imp' ('not' b)) 'imp' (b 'imp' a) = I_el Y
t32_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (('not' b) 'imp' ('not' a)) = I_el Y
t33_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' ('not' b)) 'imp' (b 'imp' ('not' a)) = I_el Y
t34_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (('not' a) 'imp' b) 'imp' (('not' b) 'imp' a) = I_el Y
t35_bvfunc_5:: for Y being non empty set for a being Function of Y,BOOLEAN holds (a 'imp' ('not' a)) 'imp' ('not' a) = I_el Y
t36_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ('not' a) 'imp' (a 'imp' b) = I_el Y
t37_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds 'not' ((a '&' b) '&' c) = (('not' a) 'or' ('not' b)) 'or' ('not' c)
t38_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds 'not' ((a 'or' b) 'or' c) = (('not' a) '&' ('not' b)) '&' ('not' c)
t39_bvfunc_5:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds a 'or' ((b '&' c) '&' d) = ((a 'or' b) '&' (a 'or' c)) '&' (a 'or' d)
t4_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN st b = I_el Y holds a 'imp' b = I_el Y
t40_bvfunc_5:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds a '&' ((b 'or' c) 'or' d) = ((a '&' b) 'or' (a '&' c)) 'or' (a '&' d)
t5_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN st 'not' a = I_el Y holds a 'imp' b = I_el Y
t6_bvfunc_5:: for Y being non empty set for a being Function of Y,BOOLEAN holds 'not' (a '&' ('not' a)) = I_el Y
t7_bvfunc_5:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' a = I_el Y
t8_bvfunc_5:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ( a 'imp' b = I_el Y iff ('not' b) 'imp' ('not' a) = I_el Y )
t9_bvfunc_5:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' b = I_el Y & b 'imp' c = I_el Y holds a 'imp' c = I_el Y
t1_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' (b 'imp' (a '&' b)) = I_el Y
t10_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' (b '&' ('not' b))) 'imp' ('not' a) = I_el Y
t11_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'or' b) '&' (a 'or' c)) 'imp' (a 'or' (b '&' c)) = I_el Y
t12_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' (b 'or' c)) 'imp' ((a '&' b) 'or' (a '&' c)) = I_el Y
t13_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'or' c) '&' (b 'or' c)) 'imp' ((a '&' b) 'or' c) = I_el Y
t14_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'or' b) '&' c) 'imp' ((a '&' c) 'or' (b '&' c)) = I_el Y
t15_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN st a '&' b = I_el Y holds a 'or' b = I_el Y
t16_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' b = I_el Y holds (a 'or' c) 'imp' (b 'or' c) = I_el Y
t17_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' b = I_el Y holds (a '&' c) 'imp' (b '&' c) = I_el Y
t18_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st c 'imp' a = I_el Y & c 'imp' b = I_el Y holds c 'imp' (a '&' b) = I_el Y
t19_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a 'imp' c = I_el Y & b 'imp' c = I_el Y holds (a 'or' b) 'imp' c = I_el Y
t2_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((b 'imp' a) 'imp' (a 'eqv' b)) = I_el Y
t20_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN st a 'or' b = I_el Y & 'not' a = I_el Y holds b = I_el Y
t21_bvfunc_6:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a 'imp' b = I_el Y & c 'imp' d = I_el Y holds (a '&' c) 'imp' (b '&' d) = I_el Y
t22_bvfunc_6:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a 'imp' b = I_el Y & c 'imp' d = I_el Y holds (a 'or' c) 'imp' (b 'or' d) = I_el Y
t23_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN st (a '&' ('not' b)) 'imp' ('not' a) = I_el Y holds a 'imp' b = I_el Y
t24_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN st a 'imp' ('not' b) = I_el Y holds b 'imp' ('not' a) = I_el Y
t25_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN st ('not' a) 'imp' b = I_el Y holds ('not' b) 'imp' a = I_el Y
t26_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'imp' (a 'or' b) = I_el Y
t27_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'or' b) 'imp' (('not' a) 'imp' b) = I_el Y
t28_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ('not' (a 'or' b)) 'imp' (('not' a) '&' ('not' b)) = I_el Y
t29_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (('not' a) '&' ('not' b)) 'imp' ('not' (a 'or' b)) = I_el Y
t3_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'or' b) 'eqv' (b 'or' a) = I_el Y
t30_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ('not' (a 'or' b)) 'imp' ('not' a) = I_el Y
t31_bvfunc_6:: for Y being non empty set for a being Function of Y,BOOLEAN holds (a 'or' a) 'imp' a = I_el Y
t32_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a '&' ('not' a)) 'imp' b = I_el Y
t33_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' (('not' a) 'or' b) = I_el Y
t34_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a '&' b) 'imp' ('not' (a 'imp' ('not' b))) = I_el Y
t35_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ('not' (a 'imp' ('not' b))) 'imp' (a '&' b) = I_el Y
t36_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ('not' (a '&' b)) 'imp' (('not' a) 'or' ('not' b)) = I_el Y
t37_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (('not' a) 'or' ('not' b)) 'imp' ('not' (a '&' b)) = I_el Y
t38_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a '&' b) 'imp' a = I_el Y
t39_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a '&' b) 'imp' (a 'or' b) = I_el Y
t4_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a '&' b) 'imp' c) 'imp' (a 'imp' (b 'imp' c)) = I_el Y
t40_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a '&' b) 'imp' b = I_el Y
t41_bvfunc_6:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' (a '&' a) = I_el Y
t42_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'eqv' b) 'imp' (a 'imp' b) = I_el Y
t43_bvfunc_6:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'eqv' b) 'imp' (b 'imp' a) = I_el Y by Th42;
t44_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c)) = I_el Y
t45_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a '&' b) '&' c) 'imp' (a '&' (b '&' c)) = I_el Y
t46_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' (b 'or' c)) 'imp' ((a 'or' b) 'or' c) = I_el Y
t5_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' (b 'imp' c)) 'imp' ((a '&' b) 'imp' c) = I_el Y
t6_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (c 'imp' a) 'imp' ((c 'imp' b) 'imp' (c 'imp' (a '&' b))) = I_el Y
t7_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'or' b) 'imp' c) 'imp' ((a 'imp' c) 'or' (b 'imp' c)) = I_el Y
t8_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' c) 'imp' ((b 'imp' c) 'imp' ((a 'or' b) 'imp' c)) = I_el Y
t9_bvfunc_6:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'imp' c) '&' (b 'imp' c)) 'imp' ((a 'or' b) 'imp' c) = I_el Y
t1_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (('not' a) 'imp' b) = b
t10_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' (a 'imp' b) = a '&' b
t11_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' ('not' b) = ('not' a) '&' ('not' b)
t12_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (b 'imp' c) = ((a 'imp' b) '&' (b 'imp' c)) '&' (a 'imp' c)
t13_bvfunc_7:: for Y being non empty set for a being Function of Y,BOOLEAN holds (I_el Y) 'imp' a = a
t14_bvfunc_7:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' (O_el Y) = 'not' a
t15_bvfunc_7:: for Y being non empty set for a being Function of Y,BOOLEAN holds (O_el Y) 'imp' a = I_el Y
t16_bvfunc_7:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' (I_el Y) = I_el Y
t17_bvfunc_7:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' ('not' a) = 'not' a
t18_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' b '<' (c 'imp' a) 'imp' (c 'imp' b)
t19_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' b '<' (a 'eqv' c) 'eqv' (b 'eqv' c)
t2_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (a 'imp' ('not' b)) = 'not' a
t20_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' b '<' (a 'imp' c) 'eqv' (b 'imp' c)
t21_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' b '<' (c 'imp' a) 'eqv' (c 'imp' b)
t22_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' b '<' (a '&' c) 'eqv' (b '&' c)
t23_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'eqv' b '<' (a 'or' c) 'eqv' (b 'or' c)
t24_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '<' ((a 'eqv' b) 'eqv' (b 'eqv' a)) 'eqv' a
t25_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '<' (a 'imp' b) 'eqv' b
t26_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '<' (b 'imp' a) 'eqv' a
t27_bvfunc_7:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '<' ((a '&' b) 'eqv' (b '&' a)) 'eqv' a
t3_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b 'or' c) = (a 'imp' b) 'or' (a 'imp' c)
t4_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b '&' c) = (a 'imp' b) '&' (a 'imp' c)
t5_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) 'imp' c = (a 'imp' c) '&' (b 'imp' c)
t6_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = (a 'imp' c) 'or' (b 'imp' c)
t7_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = a 'imp' (b 'imp' c)
t8_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'imp' c = a 'imp' (('not' b) 'or' c)
t9_bvfunc_7:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b 'or' c) = (a '&' ('not' b)) 'imp' c
t1_bvfunc_8:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds a 'imp' ((b '&' c) '&' d) = ((a 'imp' b) '&' (a 'imp' c)) '&' (a 'imp' d)
t10_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' b = a '&' (('not' a) 'or' b)
t11_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'or' b = a 'or' (('not' a) '&' b)
t12_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' b = 'not' (a 'eqv' b)
t13_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' b = (a 'or' b) '&' (('not' a) 'or' ('not' b))
t14_bvfunc_8:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'xor' (I_el Y) = 'not' a
t15_bvfunc_8:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'xor' (O_el Y) = a
t16_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'xor' b = ('not' a) 'xor' ('not' b)
t17_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'xor' b) = a 'xor' ('not' b)
t18_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a 'or' ('not' b)) '&' (('not' a) 'or' b)
t19_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a 'eqv' b = (a '&' b) 'or' (('not' a) '&' ('not' b))
t2_bvfunc_8:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds a 'imp' ((b 'or' c) 'or' d) = ((a 'imp' b) 'or' (a 'imp' c)) 'or' (a 'imp' d)
t20_bvfunc_8:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'eqv' (I_el Y) = a
t21_bvfunc_8:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'eqv' (O_el Y) = 'not' a
t22_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' (a 'eqv' b) = a 'eqv' ('not' b)
t23_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' a '<' (a 'imp' b) 'eqv' ('not' a)
t24_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds 'not' a '<' (b 'imp' a) 'eqv' ('not' b)
t25_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '<' ((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a
t26_bvfunc_8:: for Y being non empty set for a being Function of Y,BOOLEAN holds a 'imp' (('not' a) 'eqv' ('not' a)) = I_el Y
t27_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds ((a 'imp' b) 'imp' a) 'imp' a = I_el Y
t28_bvfunc_8:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds (((a 'imp' c) '&' (b 'imp' d)) '&' (('not' c) 'or' ('not' d))) 'imp' (('not' a) 'or' ('not' b)) = I_el Y
t29_bvfunc_8:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) 'imp' ((a 'imp' (b 'imp' c)) 'imp' (a 'imp' c)) = I_el Y
t3_bvfunc_8:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds ((a '&' b) '&' c) 'imp' d = ((a 'imp' d) 'or' (b 'imp' d)) 'or' (c 'imp' d)
t4_bvfunc_8:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c) 'imp' d = ((a 'imp' d) '&' (b 'imp' d)) '&' (c 'imp' d)
t5_bvfunc_8:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a) = ((((a 'imp' b) '&' (b 'imp' c)) '&' (c 'imp' a)) '&' (b 'imp' a)) '&' (a 'imp' c)
t6_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a = (a '&' b) 'or' (a '&' ('not' b))
t7_bvfunc_8:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a = (a 'or' b) '&' (a 'or' ('not' b))
t8_bvfunc_8:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a = ((((a '&' b) '&' c) 'or' ((a '&' b) '&' ('not' c))) 'or' ((a '&' ('not' b)) '&' c)) 'or' ((a '&' ('not' b)) '&' ('not' c))
t9_bvfunc_8:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a = ((((a 'or' b) 'or' c) '&' ((a 'or' b) 'or' ('not' c))) '&' ((a 'or' ('not' b)) 'or' c)) '&' ((a 'or' ('not' b)) 'or' ('not' c))
t1_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) '&' (b 'imp' c) '<' a 'or' c
t10_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' b '<' (a '&' c) 'imp' (b '&' c)
t11_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' b '<' a 'imp' (b 'or' c)
t12_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' b '<' (a 'or' c) 'imp' (b 'or' c)
t13_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a '&' b) 'or' c '<' a 'or' c
t14_bvfunc_9:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds (a '&' b) 'or' (c '&' d) '<' a 'or' c
t15_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) '&' (b 'imp' c) '<' a 'or' c by Th1;
t16_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (('not' a) 'imp' c) '<' b 'or' c
t17_bvfunc_9:: for Y being non empty set for a, c, b being Function of Y,BOOLEAN holds (a 'imp' c) '&' (b 'imp' ('not' c)) '<' ('not' a) 'or' ('not' b)
t18_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) '&' (('not' a) 'or' c) '<' b 'or' c
t19_bvfunc_9:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds (a 'imp' b) '&' (c 'imp' d) '<' (a '&' c) 'imp' (b '&' d)
t2_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' (a 'imp' b) '<' b
t20_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (a 'imp' c) '<' a 'imp' (b '&' c)
t21_bvfunc_9:: for Y being non empty set for a, c, b being Function of Y,BOOLEAN holds (a 'imp' c) '&' (b 'imp' c) '<' (a 'or' b) 'imp' c
t22_bvfunc_9:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds (a 'imp' b) '&' (c 'imp' d) '<' (a 'or' c) 'imp' (b 'or' d)
t23_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'imp' b) '&' (a 'imp' c) '<' a 'imp' (b 'or' c)
t24_bvfunc_9:: for Y being non empty set for a1, b1, c1, a2, b2, c2 being Function of Y,BOOLEAN holds ((((b1 'imp' b2) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2)) '<' a2 'imp' a1
t25_bvfunc_9:: for Y being non empty set for a1, b1, c1, a2, b2, c2 being Function of Y,BOOLEAN holds ((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2)) '<' ((a2 'imp' a1) '&' (b2 'imp' b1)) '&' (c2 'imp' c1)
t26_bvfunc_9:: for Y being non empty set for a1, b1, a2, b2 being Function of Y,BOOLEAN holds (((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ('not' (a2 '&' b2))) 'imp' ('not' (a1 '&' b1)) = I_el Y
t27_bvfunc_9:: for Y being non empty set for a1, b1, c1, a2, b2, c2 being Function of Y,BOOLEAN holds (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2)) '<' (('not' (a1 '&' b1)) '&' ('not' (a1 '&' c1))) '&' ('not' (b1 '&' c1))
t28_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' b '<' a by Lm1;
t29_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds ( (a '&' b) '&' c '<' a & (a '&' b) '&' c '<' b ) by Lm2;
t3_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' ('not' b) '<' 'not' a
t30_bvfunc_9:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN holds ( ((a '&' b) '&' c) '&' d '<' a & ((a '&' b) '&' c) '&' d '<' b ) by Lm3;
t31_bvfunc_9:: for Y being non empty set for a, b, c, d, e being Function of Y,BOOLEAN holds ( (((a '&' b) '&' c) '&' d) '&' e '<' a & (((a '&' b) '&' c) '&' d) '&' e '<' b ) by Lm4;
t32_bvfunc_9:: for Y being non empty set for a, b, c, d, e, f being Function of Y,BOOLEAN holds ( ((((a '&' b) '&' c) '&' d) '&' e) '&' f '<' a & ((((a '&' b) '&' c) '&' d) '&' e) '&' f '<' b ) by Lm5;
t33_bvfunc_9:: for Y being non empty set for a, b, c, d, e, f, g being Function of Y,BOOLEAN holds ( (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g '<' a & (((((a '&' b) '&' c) '&' d) '&' e) '&' f) '&' g '<' b ) by Lm6;
t34_bvfunc_9:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a '<' b & c '<' d holds a '&' c '<' b '&' d
t35_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN st a '&' b '<' c holds a '&' ('not' c) '<' 'not' b
t36_bvfunc_9:: for Y being non empty set for a, c, b being Function of Y,BOOLEAN holds ((a 'imp' c) '&' (b 'imp' c)) '&' (a 'or' b) '<' c
t37_bvfunc_9:: for Y being non empty set for a, c, b being Function of Y,BOOLEAN holds ((a 'imp' c) 'or' (b 'imp' c)) '&' (a '&' b) '<' c
t38_bvfunc_9:: for Y being non empty set for a, b, c, d being Function of Y,BOOLEAN st a '<' b & c '<' d holds a 'or' c '<' b 'or' d
t39_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '<' a 'or' b
t4_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'or' b) '&' ('not' a) '<' b
t40_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds a '&' b '<' a 'or' b
t5_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (('not' a) 'imp' b) '<' b
t6_bvfunc_9:: for Y being non empty set for a, b being Function of Y,BOOLEAN holds (a 'imp' b) '&' (a 'imp' ('not' b)) '<' 'not' a
t7_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' (b '&' c) '<' a 'imp' b
t8_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds (a 'or' b) 'imp' c '<' a 'imp' c
t9_bvfunc_9:: for Y being non empty set for a, b, c being Function of Y,BOOLEAN holds a 'imp' b '<' (a '&' c) 'imp' b
d1_c0sp1:: for V being non empty addLoopStr for V1 being Subset of V holds ( V1 is having-inverse iff for v being Element of V st v in V1 holds - v in V1 );
d10_c0sp1:: for V being Algebra for V1 being Subset of V holds ( V1 is additively-linearly-closed iff ( V1 is add-closed & V1 is having-inverse & ( for a being Real for v being Element of V st v in V1 holds a * v in V1 ) ) );
d11_c0sp1:: for V being Algebra for V1 being Subset of V st V1 is additively-linearly-closed & not V1 is empty holds Mult_ (V1,V) = the Mult of V | [:REAL,V1:];
d12_c0sp1:: for V being non empty RLSStruct holds ( V is scalar-mult-cancelable iff for a being Real for v being Element of V holds ( not a * v = 0. V or a = 0 or v = 0. V ) );
d13_c0sp1:: for X being non empty set holds BoundedFunctions X = { f where f is Function of X,REAL : f | X is bounded } ;
d14_c0sp1:: for X being non empty set holds R_Algebra_of_BoundedFunctions X = AlgebraStr(# (BoundedFunctions X),(mult_ ((BoundedFunctions X),(RAlgebra X))),(Add_ ((BoundedFunctions X),(RAlgebra X))),(Mult_ ((BoundedFunctions X),(RAlgebra X))),(One_ ((BoundedFunctions X),(RAlgebra X))),(Zero_ ((BoundedFunctions X),(RAlgebra X))) #);
d15_c0sp1:: for X being non empty set for F being set st F in BoundedFunctions X holds for b3 being Function of X,REAL holds ( b3 = modetrans (F,X) iff ( b3 = F & b3 | X is bounded ) );
d16_c0sp1:: for X being non empty set for f being Function of X,REAL holds PreNorms f = { (abs (f . x)) where x is Element of X : verum } ;
d17_c0sp1:: for X being non empty set for b2 being Function of (BoundedFunctions X),REAL holds ( b2 = BoundedFunctionsNorm X iff for x being set st x in BoundedFunctions X holds b2 . x = upper_bound (PreNorms (modetrans (x,X))) );
d18_c0sp1:: for X being non empty set holds R_Normed_Algebra_of_BoundedFunctions X = Normed_AlgebraStr(# (BoundedFunctions X),(mult_ ((BoundedFunctions X),(RAlgebra X))),(Add_ ((BoundedFunctions X),(RAlgebra X))),(Mult_ ((BoundedFunctions X),(RAlgebra X))),(One_ ((BoundedFunctions X),(RAlgebra X))),(Zero_ ((BoundedFunctions X),(RAlgebra X))),(BoundedFunctionsNorm X) #);
d2_c0sp1:: for V being non empty addLoopStr for V1 being Subset of V holds ( V1 is additively-closed iff ( V1 is add-closed & V1 is having-inverse ) );
d3_c0sp1:: for V, b2 being Ring holds ( b2 is Subring of V iff ( the carrier of b2 c= the carrier of V & the addF of b2 = the addF of V || the carrier of b2 & the multF of b2 = the multF of V || the carrier of b2 & 1. b2 = 1. V & 0. b2 = 0. V ) );
d4_c0sp1:: for V being non empty multLoopStr_0 for V1 being Subset of V holds ( V1 is multiplicatively-closed iff ( 1. V in V1 & ( for v, u being Element of V st v in V1 & u in V1 holds v * u in V1 ) ) );
d5_c0sp1:: for V being non empty addLoopStr for V1 being Subset of V st V1 is add-closed & not V1 is empty holds Add_ (V1,V) = the addF of V || V1;
d6_c0sp1:: for V being non empty multLoopStr_0 for V1 being Subset of V st V1 is multiplicatively-closed & not V1 is empty holds mult_ (V1,V) = the multF of V || V1;
d7_c0sp1:: for V being non empty right_complementable add-associative right_zeroed doubleLoopStr for V1 being Subset of V st V1 is add-closed & V1 is having-inverse & not V1 is empty holds Zero_ (V1,V) = 0. V;
d8_c0sp1:: for V being non empty multLoopStr_0 for V1 being Subset of V st V1 is multiplicatively-closed & not V1 is empty holds One_ (V1,V) = 1. V;
d9_c0sp1:: for V, b2 being Algebra holds ( b2 is Subalgebra of V iff ( the carrier of b2 c= the carrier of V & the addF of b2 = the addF of V || the carrier of b2 & the multF of b2 = the multF of V || the carrier of b2 & the Mult of b2 = the Mult of V | [:REAL, the carrier of b2:] & 1. b2 = 1. V & 0. b2 = 0. V ) );
t1_c0sp1:: for X being non empty set for d1, d2 being Element of X for A being BinOp of X for M being Function of [:X,X:],X for V being Ring for V1 being Subset of V st V1 = X & A = the addF of V || V1 & M = the multF of V || V1 & d1 = 1. V & d2 = 0. V & V1 is having-inverse holds doubleLoopStr(# X,A,M,d1,d2 #) is Subring of V
t10_c0sp1:: for X being non empty set holds R_Algebra_of_BoundedFunctions X is Subalgebra of RAlgebra X by Th6;
t11_c0sp1:: for X being non empty set holds R_Algebra_of_BoundedFunctions X is RealLinearSpace
t12_c0sp1:: for X being non empty set for F, G, H being VECTOR of (R_Algebra_of_BoundedFunctions X) for f, g, h being Function of X,REAL st f = F & g = G & h = H holds ( H = F + G iff for x being Element of X holds h . x = (f . x) + (g . x) )
t13_c0sp1:: for X being non empty set for a being Real for F, G being VECTOR of (R_Algebra_of_BoundedFunctions X) for f, g being Function of X,REAL st f = F & g = G holds ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
t14_c0sp1:: for X being non empty set for F, G, H being VECTOR of (R_Algebra_of_BoundedFunctions X) for f, g, h being Function of X,REAL st f = F & g = G & h = H holds ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )
t15_c0sp1:: for X being non empty set holds 0. (R_Algebra_of_BoundedFunctions X) = X --> 0
t16_c0sp1:: for X being non empty set holds 1_ (R_Algebra_of_BoundedFunctions X) = X --> 1
t17_c0sp1:: for X being non empty set for f being Function of X,REAL st f | X is bounded holds PreNorms f is bounded_above
t18_c0sp1:: for X being non empty set for f being Function of X,REAL holds ( f | X is bounded iff PreNorms f is bounded_above )
t19_c0sp1:: for X being non empty set for f being Function of X,REAL st f | X is bounded holds modetrans (f,X) = f
t2_c0sp1:: for V being Ring for V1 being Subset of V st V1 is additively-closed & V1 is multiplicatively-closed & not V1 is empty holds doubleLoopStr(# V1,(Add_ (V1,V)),(mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is Ring
t20_c0sp1:: for X being non empty set for f being Function of X,REAL st f | X is bounded holds (BoundedFunctionsNorm X) . f = upper_bound (PreNorms f)
t21_c0sp1:: for W being Normed_AlgebraStr for V being Algebra st AlgebraStr(# the carrier of W, the multF of W, the addF of W, the Mult of W, the OneF of W, the ZeroF of W #) = V holds W is Algebra
t22_c0sp1:: for X being non empty set holds R_Normed_Algebra_of_BoundedFunctions X is Algebra
t23_c0sp1:: for X being non empty set for F being Point of (R_Normed_Algebra_of_BoundedFunctions X) holds (Mult_ ((BoundedFunctions X),(RAlgebra X))) . (1,F) = F
t24_c0sp1:: for X being non empty set holds R_Normed_Algebra_of_BoundedFunctions X is RealLinearSpace
t25_c0sp1:: for X being non empty set holds X --> 0 = 0. (R_Normed_Algebra_of_BoundedFunctions X)
t26_c0sp1:: for X being non empty set for x being Element of X for f being Function of X,REAL for F being Point of (R_Normed_Algebra_of_BoundedFunctions X) st f = F & f | X is bounded holds abs (f . x) <= ||.F.||
t27_c0sp1:: for X being non empty set for F being Point of (R_Normed_Algebra_of_BoundedFunctions X) holds 0 <= ||.F.||
t28_c0sp1:: for X being non empty set for F being Point of (R_Normed_Algebra_of_BoundedFunctions X) st F = 0. (R_Normed_Algebra_of_BoundedFunctions X) holds 0 = ||.F.||
t29_c0sp1:: for X being non empty set for f, g, h being Function of X,REAL for F, G, H being Point of (R_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G & h = H holds ( H = F + G iff for x being Element of X holds h . x = (f . x) + (g . x) )
t3_c0sp1:: for X being non empty set for d1, d2 being Element of X for A being BinOp of X for M being Function of [:X,X:],X for V being Algebra for V1 being Subset of V for MR being Function of [:REAL,X:],X st V1 = X & d1 = 0. V & d2 = 1. V & A = the addF of V || V1 & M = the multF of V || V1 & MR = the Mult of V | [:REAL,V1:] & V1 is having-inverse holds AlgebraStr(# X,M,A,MR,d2,d1 #) is Subalgebra of V
t30_c0sp1:: for X being non empty set for a being Real for f, g being Function of X,REAL for F, G being Point of (R_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G holds ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
t31_c0sp1:: for X being non empty set for f, g, h being Function of X,REAL for F, G, H being Point of (R_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G & h = H holds ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )
t32_c0sp1:: for X being non empty set for a being Real for F, G being Point of (R_Normed_Algebra_of_BoundedFunctions X) holds ( ( ||.F.|| = 0 implies F = 0. (R_Normed_Algebra_of_BoundedFunctions X) ) & ( F = 0. (R_Normed_Algebra_of_BoundedFunctions X) implies ||.F.|| = 0 ) & ||.(a * F).|| = (abs a) * ||.F.|| & ||.(F + G).|| <= ||.F.|| + ||.G.|| )
t33_c0sp1:: for X being non empty set holds ( R_Normed_Algebra_of_BoundedFunctions X is reflexive & R_Normed_Algebra_of_BoundedFunctions X is discerning & R_Normed_Algebra_of_BoundedFunctions X is RealNormSpace-like )
t34_c0sp1:: for X being non empty set for f, g, h being Function of X,REAL for F, G, H being Point of (R_Normed_Algebra_of_BoundedFunctions X) st f = F & g = G & h = H holds ( H = F - G iff for x being Element of X holds h . x = (f . x) - (g . x) )
t35_c0sp1:: for X being non empty set for seq being sequence of (R_Normed_Algebra_of_BoundedFunctions X) st seq is Cauchy_sequence_by_Norm holds seq is convergent
t36_c0sp1:: for X being non empty set holds R_Normed_Algebra_of_BoundedFunctions X is RealBanachSpace
t37_c0sp1:: for X being non empty set holds R_Normed_Algebra_of_BoundedFunctions X is Banach_Algebra
t4_c0sp1:: for V being non empty right_complementable add-associative right_zeroed vector-distributive scalar-distributive scalar-associative vector-associative AlgebraStr for a being Real holds a * (0. V) = 0. V
t5_c0sp1:: for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative vector-associative AlgebraStr st V is scalar-mult-cancelable holds V is RealLinearSpace
t6_c0sp1:: for V being Algebra for V1 being Subset of V st V1 is additively-linearly-closed & V1 is multiplicatively-closed & not V1 is empty holds AlgebraStr(# V1,(mult_ (V1,V)),(Add_ (V1,V)),(Mult_ (V1,V)),(One_ (V1,V)),(Zero_ (V1,V)) #) is Subalgebra of V
t7_c0sp1:: for X being non empty set holds RAlgebra X is RealLinearSpace
t8_c0sp1:: for V being Algebra for V1 being Subalgebra of V holds ( ( for v1, w1 being Element of V1 for v, w being Element of V st v1 = v & w1 = w holds v1 + w1 = v + w ) & ( for v1, w1 being Element of V1 for v, w being Element of V st v1 = v & w1 = w holds v1 * w1 = v * w ) & ( for v1 being Element of V1 for v being Element of V for a being Real st v1 = v holds a * v1 = a * v ) & 1_ V1 = 1_ V & 0. V1 = 0. V )
t9_c0sp1:: for X being non empty set holds ( BoundedFunctions X is additively-linearly-closed & BoundedFunctions X is multiplicatively-closed )
d1_c0sp2:: for X being 1-sorted for y being real number holds X --> y = the carrier of X --> y;
d2_c0sp2:: for X being non empty TopSpace holds ContinuousFunctions X = { f where f is continuous RealMap of X : verum } ;
d3_c0sp2:: for X being non empty TopSpace holds R_Algebra_of_ContinuousFunctions X = AlgebraStr(# (ContinuousFunctions X),(mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Add_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(One_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Zero_ ((ContinuousFunctions X),(RAlgebra the carrier of X))) #);
d4_c0sp2:: for X being non empty compact TopSpace holds ContinuousFunctionsNorm X = (BoundedFunctionsNorm the carrier of X) | (ContinuousFunctions X);
d5_c0sp2:: for X being non empty compact TopSpace holds R_Normed_Algebra_of_ContinuousFunctions X = Normed_AlgebraStr(# (ContinuousFunctions X),(mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Add_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(One_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(Zero_ ((ContinuousFunctions X),(RAlgebra the carrier of X))),(ContinuousFunctionsNorm X) #);
d6_c0sp2:: for X being non empty TopSpace holds C_0_Functions X = { f where f is RealMap of X : ( f is continuous & ex Y being non empty Subset of X st ( Y is compact & ( for A being Subset of X st A = support f holds Cl A is Subset of Y ) ) ) } ;
d7_c0sp2:: for X being non empty TopSpace holds R_VectorSpace_of_C_0_Functions X = RLSStruct(# (C_0_Functions X),(Zero_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Add_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Mult_ ((C_0_Functions X),(RealVectSpace the carrier of X))) #);
d8_c0sp2:: for X being non empty TopSpace holds C_0_FunctionsNorm X = (BoundedFunctionsNorm the carrier of X) | (C_0_Functions X);
d9_c0sp2:: for X being non empty TopSpace holds R_Normed_Space_of_C_0_Functions X = NORMSTR(# (C_0_Functions X),(Zero_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Add_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(Mult_ ((C_0_Functions X),(RealVectSpace the carrier of X))),(C_0_FunctionsNorm X) #);
t1_c0sp2:: for X being non empty TopSpace for f being RealMap of X holds ( f is continuous iff for x being Point of X for V being Subset of REAL st f . x in V & V is open holds ex W being Subset of X st ( x in W & W is open & f .: W c= V ) )
t10_c0sp2:: for W being Normed_AlgebraStr for V being Algebra st AlgebraStr(# the carrier of W, the multF of W, the addF of W, the Mult of W, the OneF of W, the ZeroF of W #) = V holds W is Algebra
t11_c0sp2:: for X being non empty compact TopSpace for F being Point of (R_Normed_Algebra_of_ContinuousFunctions X) holds (Mult_ ((ContinuousFunctions X),(RAlgebra the carrier of X))) . (1,F) = F
t12_c0sp2:: for X being non empty compact TopSpace holds X --> 0 = 0. (R_Normed_Algebra_of_ContinuousFunctions X)
t13_c0sp2:: for X being non empty compact TopSpace for F being Point of (R_Normed_Algebra_of_ContinuousFunctions X) holds 0 <= ||.F.||
t14_c0sp2:: for X being non empty compact TopSpace for F being Point of (R_Normed_Algebra_of_ContinuousFunctions X) st F = 0. (R_Normed_Algebra_of_ContinuousFunctions X) holds 0 = ||.F.||
t15_c0sp2:: for X being non empty compact TopSpace for F, G, H being Point of (R_Normed_Algebra_of_ContinuousFunctions X) for f, g, h being RealMap of X st f = F & g = G & h = H holds ( H = F + G iff for x being Element of X holds h . x = (f . x) + (g . x) )
t16_c0sp2:: for a being Real for X being non empty compact TopSpace for F, G being Point of (R_Normed_Algebra_of_ContinuousFunctions X) for f, g being RealMap of X st f = F & g = G holds ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
t17_c0sp2:: for X being non empty compact TopSpace for F, G, H being Point of (R_Normed_Algebra_of_ContinuousFunctions X) for f, g, h being RealMap of X st f = F & g = G & h = H holds ( H = F * G iff for x being Element of X holds h . x = (f . x) * (g . x) )
t18_c0sp2:: for a being Real for X being non empty compact TopSpace for F, G being Point of (R_Normed_Algebra_of_ContinuousFunctions X) holds ( ( ||.F.|| = 0 implies F = 0. (R_Normed_Algebra_of_ContinuousFunctions X) ) & ( F = 0. (R_Normed_Algebra_of_ContinuousFunctions X) implies ||.F.|| = 0 ) & ||.(a * F).|| = (abs a) * ||.F.|| & ||.(F + G).|| <= ||.F.|| + ||.G.|| )
t19_c0sp2:: for X being non empty compact TopSpace for F, G, H being Point of (R_Normed_Algebra_of_ContinuousFunctions X) for f, g, h being RealMap of X st f = F & g = G & h = H holds ( H = F - G iff for x being Element of X holds h . x = (f . x) - (g . x) )
t2_c0sp2:: for X being non empty TopSpace holds R_Algebra_of_ContinuousFunctions X is Subalgebra of RAlgebra the carrier of X by C0SP1:6;
t20_c0sp2:: for X being RealBanachSpace for Y being Subset of X for seq being sequence of X st Y is closed & rng seq c= Y & seq is Cauchy_sequence_by_Norm holds ( seq is convergent & lim seq in Y )
t21_c0sp2:: for X being non empty compact TopSpace for Y being Subset of (R_Normed_Algebra_of_BoundedFunctions the carrier of X) st Y = ContinuousFunctions X holds Y is closed
t22_c0sp2:: for X being non empty compact TopSpace for seq being sequence of (R_Normed_Algebra_of_ContinuousFunctions X) st seq is Cauchy_sequence_by_Norm holds seq is convergent
t23_c0sp2:: for X being non empty TopSpace for f, g being RealMap of X holds support (f + g) c= (support f) \/ (support g)
t24_c0sp2:: for X being non empty TopSpace for a being Real for f being RealMap of X holds support (a (#) f) c= support f
t25_c0sp2:: for X being non empty TopSpace for f, g being RealMap of X holds support (f (#) g) c= (support f) \/ (support g)
t26_c0sp2:: for X being non empty TopSpace holds C_0_Functions X is non empty Subset of (RAlgebra the carrier of X) ;
t27_c0sp2:: for X being non empty TopSpace for W being non empty Subset of (RAlgebra the carrier of X) st W = C_0_Functions X holds W is additively-linearly-closed
t28_c0sp2:: for X being non empty TopSpace holds C_0_Functions X is linearly-closed
t29_c0sp2:: for X being non empty TopSpace holds R_VectorSpace_of_C_0_Functions X is Subspace of RealVectSpace the carrier of X by RSSPACE:11;
t3_c0sp2:: for X being non empty TopSpace for F, G, H being VECTOR of (R_Algebra_of_ContinuousFunctions X) for f, g, h being RealMap of X st f = F & g = G & h = H holds ( H = F + G iff for x being Element of the carrier of X holds h . x = (f . x) + (g . x) )
t30_c0sp2:: for X being non empty TopSpace for x being set st x in C_0_Functions X holds x in BoundedFunctions the carrier of X
t31_c0sp2:: for X being non empty TopSpace for x being set st x in C_0_Functions X holds x in ContinuousFunctions X
t32_c0sp2:: for X being non empty TopSpace holds 0. (R_VectorSpace_of_C_0_Functions X) = X --> 0
t33_c0sp2:: for X being non empty TopSpace holds 0. (R_Normed_Space_of_C_0_Functions X) = X --> 0
t34_c0sp2:: for a being Real for X being non empty TopSpace for F, G being Point of (R_Normed_Space_of_C_0_Functions X) holds ( ( ||.F.|| = 0 implies F = 0. (R_Normed_Space_of_C_0_Functions X) ) & ( F = 0. (R_Normed_Space_of_C_0_Functions X) implies ||.F.|| = 0 ) & ||.(a * F).|| = (abs a) * ||.F.|| & ||.(F + G).|| <= ||.F.|| + ||.G.|| )
t35_c0sp2:: for X being non empty TopSpace holds R_Normed_Space_of_C_0_Functions X is RealNormSpace-like
t36_c0sp2:: for X being non empty TopSpace holds R_Normed_Space_of_C_0_Functions X is RealNormSpace ;
t4_c0sp2:: for X being non empty TopSpace for F, G, H being VECTOR of (R_Algebra_of_ContinuousFunctions X) for f, g, h being RealMap of X for a being Real st f = F & g = G holds ( G = a * F iff for x being Element of X holds g . x = a * (f . x) )
t5_c0sp2:: for X being non empty TopSpace for F, G, H being VECTOR of (R_Algebra_of_ContinuousFunctions X) for f, g, h being RealMap of X st f = F & g = G & h = H holds ( H = F * G iff for x being Element of the carrier of X holds h . x = (f . x) * (g . x) )
t6_c0sp2:: for X being non empty TopSpace holds 0. (R_Algebra_of_ContinuousFunctions X) = X --> 0
t7_c0sp2:: for X being non empty TopSpace holds 1_ (R_Algebra_of_ContinuousFunctions X) = X --> 1
t8_c0sp2:: for A being Algebra for A1, A2 being Subalgebra of A st the carrier of A1 c= the carrier of A2 holds A1 is Subalgebra of A2
t9_c0sp2:: for X being non empty compact TopSpace holds R_Algebra_of_ContinuousFunctions X is Subalgebra of R_Algebra_of_BoundedFunctions the carrier of X
d1_calcul_1:: for D being non empty set for f, b3 being FinSequence of D holds ( ( len f > 0 implies ( b3 = Ant f iff for i being Element of NAT st len f = i + 1 holds b3 = f | (Seg i) ) ) & ( not len f > 0 implies ( b3 = Ant f iff b3 = {} ) ) );
d10_calcul_1:: for Al being QC-alphabet for p being Element of CQC-WFF Al for X being Subset of (CQC-WFF Al) holds ( p is_formal_provable_from X iff ex f being FinSequence of CQC-WFF Al st ( rng (Ant f) c= X & Suc f = p & |- f ) );
d11_calcul_1:: for Al being QC-alphabet for X being Subset of (CQC-WFF Al) for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= X iff for p being Element of CQC-WFF Al st p in X holds J,v |= p );
d12_calcul_1:: for Al being QC-alphabet for X being Subset of (CQC-WFF Al) for p being Element of CQC-WFF Al holds ( X |= p iff for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= X holds J,v |= p );
d13_calcul_1:: for Al being QC-alphabet for p being Element of CQC-WFF Al holds ( |= p iff {} (CQC-WFF Al) |= p );
d14_calcul_1:: for Al being QC-alphabet for f being FinSequence of CQC-WFF Al for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) holds ( J,v |= f iff J,v |= rng f );
d15_calcul_1:: for Al being QC-alphabet for f being FinSequence of CQC-WFF Al for p being Element of CQC-WFF Al holds ( f |= p iff for A being non empty set for J being interpretation of Al,A for v being Element of Valuations_in (Al,A) st J,v |= f holds J,v |= p );
d16_calcul_1:: for f being FinSequence for p being set holds ( p is_tail_of f iff ex i being Element of NAT st ( i in dom f & f . i = p ) );
d2_calcul_1:: for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds ( ( len f > 0 implies Suc f = f . (len f) ) & ( not len f > 0 implies Suc f = VERUM Al ) );
d3_calcul_1:: for f being Relation for p being set holds ( p is_tail_of f iff p in rng f );
d4_calcul_1:: for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al holds ( f is_Subsequence_of g iff ex N being Subset of NAT st f c= Seq (g | N) );
d5_calcul_1:: for Al being QC-alphabet for f being FinSequence of CQC-WFF Al for b3 being Subset of (bound_QC-variables Al) holds ( b3 = still_not-bound_in f iff for a being set holds ( a in b3 iff ex i being Element of NAT ex p being Element of CQC-WFF Al st ( i in dom f & p = f . i & a in still_not-bound_in p ) ) );
d6_calcul_1:: for Al being QC-alphabet for b2 being set holds ( b2 = set_of_CQC-WFF-seq Al iff for a being set holds ( a in b2 iff a is FinSequence of CQC-WFF Al ) );
d7_calcul_1:: for Al being QC-alphabet for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] for n being Nat holds ( ( (PR . n) `2 = 0 implies ( PR,n is_a_correct_step iff ex f being FinSequence of CQC-WFF Al st ( Suc f is_tail_of Ant f & (PR . n) `1 = f ) ) ) & ( (PR . n) `2 = 1 implies ( PR,n is_a_correct_step iff ex f being FinSequence of CQC-WFF Al st (PR . n) `1 = f ^ <*(VERUM Al)*> ) ) & ( (PR . n) `2 = 2 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) ) ) & ( (PR . n) `2 = 3 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF Al st ( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 7 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p, q being Element of CQC-WFF Al st ( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 8 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 9 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF Al ex p being Element of CQC-WFF Al ex x, y being bound_QC-variable of Al st ( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) );
d8_calcul_1:: for Al being QC-alphabet for PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] holds ( PR is a_proof iff ( PR <> {} & ( for n being Nat st 1 <= n & n <= len PR holds PR,n is_a_correct_step ) ) );
d9_calcul_1:: for Al being QC-alphabet for f being FinSequence of CQC-WFF Al holds ( |- f iff ex PR being FinSequence of [:(set_of_CQC-WFF-seq Al),Proof_Step_Kinds:] st ( PR is a_proof & f = (PR . (len PR)) `1 ) );
t1_calcul_1:: for Al being QC-alphabet for f, g being FinSequence of CQC-WFF Al st f is_Subsequence_of g holds ( rng f c= rng g & ex N being Subset of NAT st rng f c= rng (g | N) )
t10_calcul_1:: for b being set for fin being FinSequence holds ( 1 <= len (fin ^ <*b*>) & len (fin ^ <*b*>) in dom (fin ^ <*b*>) )
t11_calcul_1:: for m, n being E*