:: QMAX_1 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

definition
let X be non empty set ;
let S be SigmaField of X;
func Probabilities S -> set means :Def1: :: QMAX_1:def 1
for x being set holds
( x in it iff x is Probability of S );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is Probability of S )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is Probability of S ) ) & ( for x being set holds
( x in b2 iff x is Probability of S ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines Probabilities QMAX_1:def 1 :
for X being non empty set
for S being SigmaField of X
for b3 being set holds
( b3 = Probabilities S iff for x being set holds
( x in b3 iff x is Probability of S ) );

registration
let X be non empty set ;
let S be SigmaField of X;
cluster Probabilities S -> non empty ;
coherence
not Probabilities S is empty
proof end;
end;

definition
attr c1 is strict;
struct QM_Str -> ;
aggr QM_Str(# Observables, States, Quantum_Probability #) -> QM_Str ;
sel Observables c1 -> non empty set ;
sel States c1 -> non empty set ;
sel Quantum_Probability c1 -> Function of [:the Observables of c1,the States of c1:], Probabilities Borel_Sets ;
end;

definition
let Q be QM_Str ;
func Obs Q -> set equals :: QMAX_1:def 2
the Observables of Q;
coherence
the Observables of Q is set ;
func Sts Q -> set equals :: QMAX_1:def 3
the States of Q;
coherence
the States of Q is set ;
end;

:: deftheorem defines Obs QMAX_1:def 2 :
for Q being QM_Str holds Obs Q = the Observables of Q;

:: deftheorem defines Sts QMAX_1:def 3 :
for Q being QM_Str holds Sts Q = the States of Q;

registration
let Q be QM_Str ;
cluster Obs Q -> non empty ;
coherence
not Obs Q is empty ;
cluster Sts Q -> non empty ;
coherence
not Sts Q is empty ;
end;

definition
let Q be QM_Str ;
let A1 be Element of Obs Q;
let s be Element of Sts Q;
func Meas A1,s -> Probability of Borel_Sets equals :: QMAX_1:def 4
the Quantum_Probability of Q . [A1,s];
coherence
the Quantum_Probability of Q . [A1,s] is Probability of Borel_Sets
proof end;
end;

:: deftheorem defines Meas QMAX_1:def 4 :
for Q being QM_Str
for A1 being Element of Obs Q
for s being Element of Sts Q holds Meas A1,s = the Quantum_Probability of Q . [A1,s];

reconsider X = {0} as non empty set ;

consider P being Function of Borel_Sets , REAL such that
Lm1: for D being Subset of REAL st D in Borel_Sets holds
( ( 0 in D implies P . D = 1 ) & ( not 0 in D implies P . D = 0 ) ) by PROB_1:60;

Lm2: for A being Event of Borel_Sets holds
( ( 0 in A implies P . A = 1 ) & ( not 0 in A implies P . A = 0 ) )
proof end;

Lm3: for A being Event of Borel_Sets holds 0 <= P . A
proof end;

Lm4: P . REAL = 1
proof end;

Lm5: for A, B being Event of Borel_Sets st A misses B holds
P . (A \/ B) = (P . A) + (P . B)
proof end;

for ASeq being SetSequence of Borel_Sets st ASeq is non-increasing holds
( P * ASeq is convergent & lim (P * ASeq) = P . (Intersection ASeq) )
proof end;

then reconsider P = P as Probability of Borel_Sets by Lm3, Lm4, Lm5, PROB_1:def 13;

reconsider f = {[[0,0],P]} as Function by GRFUNC_1:15;

Lm6: ( dom f = {[0,0]} & rng f = {P} )
by RELAT_1:23;

then Lm7: dom f = [:X,X:]
by ZFMISC_1:35;

P in Probabilities Borel_Sets
by Def1;

then rng f c= Probabilities Borel_Sets
by Lm6, ZFMISC_1:37;

then reconsider Y = f as Function of [:X,X:], Probabilities Borel_Sets by Lm7, FUNCT_2:def 1, RELSET_1:11;

Lm8: now
thus for A1, A2 being Element of Obs QM_Str(# X,X,Y #) st ( for s being Element of Sts QM_Str(# X,X,Y #) holds Meas A1,s = Meas A2,s ) holds
A1 = A2 :: thesis: ( ( for s1, s2 being Element of Sts QM_Str(# X,X,Y #) st ( for A being Element of Obs QM_Str(# X,X,Y #) holds Meas A,s1 = Meas A,s2 ) holds
s1 = s2 ) & ( for s1, s2 being Element of Sts QM_Str(# X,X,Y #)
for t being Real st 0 <= t & t <= 1 holds
ex s being Element of Sts QM_Str(# X,X,Y #) st
for A being Element of Obs QM_Str(# X,X,Y #)
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) ) )
proof
let A1, A2 be Element of Obs QM_Str(# X,X,Y #); :: thesis: ( ( for s being Element of Sts QM_Str(# X,X,Y #) holds Meas A1,s = Meas A2,s ) implies A1 = A2 )
( A1 = 0 & A2 = 0 ) by TARSKI:def 1;
hence ( ( for s being Element of Sts QM_Str(# X,X,Y #) holds Meas A1,s = Meas A2,s ) implies A1 = A2 ) ; :: thesis: verum
end;
thus for s1, s2 being Element of Sts QM_Str(# X,X,Y #) st ( for A being Element of Obs QM_Str(# X,X,Y #) holds Meas A,s1 = Meas A,s2 ) holds
s1 = s2 :: thesis: for s1, s2 being Element of Sts QM_Str(# X,X,Y #)
for t being Real st 0 <= t & t <= 1 holds
ex s being Element of Sts QM_Str(# X,X,Y #) st
for A being Element of Obs QM_Str(# X,X,Y #)
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
proof
let s1, s2 be Element of Sts QM_Str(# X,X,Y #); :: thesis: ( ( for A being Element of Obs QM_Str(# X,X,Y #) holds Meas A,s1 = Meas A,s2 ) implies s1 = s2 )
( s1 = 0 & s2 = 0 ) by TARSKI:def 1;
hence ( ( for A being Element of Obs QM_Str(# X,X,Y #) holds Meas A,s1 = Meas A,s2 ) implies s1 = s2 ) ; :: thesis: verum
end;
thus for s1, s2 being Element of Sts QM_Str(# X,X,Y #)
for t being Real st 0 <= t & t <= 1 holds
ex s being Element of Sts QM_Str(# X,X,Y #) st
for A being Element of Obs QM_Str(# X,X,Y #)
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) :: thesis: verum
proof
let s1, s2 be Element of Sts QM_Str(# X,X,Y #); :: thesis: for t being Real st 0 <= t & t <= 1 holds
ex s being Element of Sts QM_Str(# X,X,Y #) st
for A being Element of Obs QM_Str(# X,X,Y #)
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
A1: ( s1 = 0 & s2 = 0 ) by TARSKI:def 1;
let t be Real; :: thesis: ( 0 <= t & t <= 1 implies ex s being Element of Sts QM_Str(# X,X,Y #) st
for A being Element of Obs QM_Str(# X,X,Y #)
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) )
assume ( 0 <= t & t <= 1 ) ; :: thesis: ex s being Element of Sts QM_Str(# X,X,Y #) st
for A being Element of Obs QM_Str(# X,X,Y #)
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
take s2 ; :: thesis: for A being Element of Obs QM_Str(# X,X,Y #)
for E being Event of Borel_Sets holds (Meas A,s2) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
let A be Element of Obs QM_Str(# X,X,Y #); :: thesis: for E being Event of Borel_Sets holds (Meas A,s2) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
let E be Event of Borel_Sets ; :: thesis: (Meas A,s2) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E))
thus (Meas A,s2) . E = (t + (1 - t)) * ((Meas A,s2) . E)
.= (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) by A1, XCMPLX_1:8 ; :: thesis: verum
end;
end;

definition
let IT be QM_Str ;
attr IT is Quantum_Mechanics-like means :Def5: :: QMAX_1:def 5
( ( for A1, A2 being Element of Obs IT st ( for s being Element of Sts IT holds Meas A1,s = Meas A2,s ) holds
A1 = A2 ) & ( for s1, s2 being Element of Sts IT st ( for A being Element of Obs IT holds Meas A,s1 = Meas A,s2 ) holds
s1 = s2 ) & ( for s1, s2 being Element of Sts IT
for t being Real st 0 <= t & t <= 1 holds
ex s being Element of Sts IT st
for A being Element of Obs IT
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) ) );
end;

:: deftheorem Def5 defines Quantum_Mechanics-like QMAX_1:def 5 :
for IT being QM_Str holds
( IT is Quantum_Mechanics-like iff ( ( for A1, A2 being Element of Obs IT st ( for s being Element of Sts IT holds Meas A1,s = Meas A2,s ) holds
A1 = A2 ) & ( for s1, s2 being Element of Sts IT st ( for A being Element of Obs IT holds Meas A,s1 = Meas A,s2 ) holds
s1 = s2 ) & ( for s1, s2 being Element of Sts IT
for t being Real st 0 <= t & t <= 1 holds
ex s being Element of Sts IT st
for A being Element of Obs IT
for E being Event of Borel_Sets holds (Meas A,s) . E = (t * ((Meas A,s1) . E)) + ((1 - t) * ((Meas A,s2) . E)) ) ) );

registration
cluster strict Quantum_Mechanics-like QM_Str ;
existence
ex b1 being QM_Str st
( b1 is strict & b1 is Quantum_Mechanics-like )
proof end;
end;

definition
mode Quantum_Mechanics is Quantum_Mechanics-like QM_Str ;
end;

definition
let X be set ;
attr c2 is strict;
struct POI_Str of X -> ;
aggr POI_Str(# Ordering, Involution #) -> POI_Str of X;
sel Ordering c2 -> Relation of X;
sel Involution c2 -> Function of X,X;
end;

definition
let X1 be set ;
let Inv be Function of X1,X1;
pred Inv is_an_involution_in X1 means :: QMAX_1:def 6
for x1 being Element of X1 holds Inv . (Inv . x1) = x1;
end;

:: deftheorem defines is_an_involution_in QMAX_1:def 6 :
for X1 being set
for Inv being Function of X1,X1 holds
( Inv is_an_involution_in X1 iff for x1 being Element of X1 holds Inv . (Inv . x1) = x1 );

definition
let X1 be set ;
let W be POI_Str of X1;
pred W is_a_Quantuum_Logic_on X1 means :Def7: :: QMAX_1:def 7
 ex Ord being Relation of X1 ex Inv being Function of X1,X1 st
( W = POI_Str(# Ord,Inv #) & Ord partially_orders X1 & Inv is_an_involution_in X1 & ( for x, y being Element of X1 st [x,y] in Ord holds
[(Inv . y),(Inv . x)] in Ord ) );
end;

:: deftheorem Def7 defines is_a_Quantuum_Logic_on QMAX_1:def 7 :
for X1 being set
for W being POI_Str of X1 holds
( W is_a_Quantuum_Logic_on X1 iff ex Ord being Relation of X1 ex Inv being Function of X1,X1 st
( W = POI_Str(# Ord,Inv #) & Ord partially_orders X1 & Inv is_an_involution_in X1 & ( for x, y being Element of X1 st [x,y] in Ord holds
[(Inv . y),(Inv . x)] in Ord ) ) );

definition
let Q be Quantum_Mechanics;
func Prop Q -> set equals :: QMAX_1:def 8
[:(Obs Q),Borel_Sets :];
coherence
[:(Obs Q),Borel_Sets :] is set ;
end;

:: deftheorem defines Prop QMAX_1:def 8 :
for Q being Quantum_Mechanics holds Prop Q = [:(Obs Q),Borel_Sets :];

registration
let Q be Quantum_Mechanics;
cluster Prop Q -> non empty ;
coherence
not Prop Q is empty ;
end;

definition
let Q be Quantum_Mechanics;
let p be Element of Prop Q;
:: original: `1
redefine func p `1 -> Element of Obs Q;
coherence
p `1 is Element of Obs Q by MCART_1:10;
:: original: `2
redefine func p `2 -> Event of Borel_Sets ;
coherence
p `2 is Event of Borel_Sets
proof end;
end;

theorem :: QMAX_1:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th14: :: QMAX_1:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p being Element of Prop Q holds p = [(p `1 ),(p `2 )] by MCART_1:23;

theorem :: QMAX_1:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th16: :: QMAX_1:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for s being Element of Sts Q
for p being Element of Prop Q
for E being Event of Borel_Sets st E = (p `2 ) ` holds
(Meas (p `1 ),s) . (p `2 ) = 1 - ((Meas (p `1 ),s) . E)
proof end;

definition
let Q be Quantum_Mechanics;
let p be Element of Prop Q;
func 'not' p -> Element of Prop Q equals :: QMAX_1:def 9
[(p `1 ),((p `2 ) ` )];
coherence
[(p `1 ),((p `2 ) ` )] is Element of Prop Q
proof end;
end;

:: deftheorem defines 'not' QMAX_1:def 9 :
for Q being Quantum_Mechanics
for p being Element of Prop Q holds 'not' p = [(p `1 ),((p `2 ) ` )];

definition
let Q be Quantum_Mechanics;
let p, q be Element of Prop Q;
pred p |- q means :Def10: :: QMAX_1:def 10
for s being Element of Sts Q holds (Meas (p `1 ),s) . (p `2 ) <= (Meas (q `1 ),s) . (q `2 );
end;

:: deftheorem Def10 defines |- QMAX_1:def 10 :
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p |- q iff for s being Element of Sts Q holds (Meas (p `1 ),s) . (p `2 ) <= (Meas (q `1 ),s) . (q `2 ) );

definition
let Q be Quantum_Mechanics;
let p, q be Element of Prop Q;
pred p <==> q means :Def11: :: QMAX_1:def 11
( p |- q & q |- p );
end;

:: deftheorem Def11 defines <==> QMAX_1:def 11 :
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p <==> q iff ( p |- q & q |- p ) );

theorem :: QMAX_1:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th20: :: QMAX_1:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p <==> q iff for s being Element of Sts Q holds (Meas (p `1 ),s) . (p `2 ) = (Meas (q `1 ),s) . (q `2 ) )
proof end;

theorem Th21: :: QMAX_1:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p being Element of Prop Q holds p |- p
proof end;

theorem Th22: :: QMAX_1:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p, q, r being Element of Prop Q st p |- q & q |- r holds
p |- r
proof end;

theorem Th23: :: QMAX_1:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p being Element of Prop Q holds p <==> p
proof end;

theorem Th24: :: QMAX_1:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p, q being Element of Prop Q st p <==> q holds
q <==> p
proof end;

theorem Th25: :: QMAX_1:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p, q, r being Element of Prop Q st p <==> q & q <==> r holds
p <==> r
proof end;

theorem Th26: :: QMAX_1:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p being Element of Prop Q holds
( ('not' p) `1 = p `1 & ('not' p) `2 = (p `2 ) ` ) by MCART_1:7;

theorem Th27: :: QMAX_1:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p being Element of Prop Q holds 'not' ('not' p) = p
proof end;

theorem Th28: :: QMAX_1:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p, q being Element of Prop Q st p |- q holds
'not' q |- 'not' p
proof end;

definition
let Q be Quantum_Mechanics;
func PropRel Q -> Equivalence_Relation of Prop Q means :Def12: :: QMAX_1:def 12
for p, q being Element of Prop Q holds
( [p,q] in it iff p <==> q );
existence
ex b1 being Equivalence_Relation of Prop Q st
for p, q being Element of Prop Q holds
( [p,q] in b1 iff p <==> q )
proof end;
uniqueness
for b1, b2 being Equivalence_Relation of Prop Q st ( for p, q being Element of Prop Q holds
( [p,q] in b1 iff p <==> q ) ) & ( for p, q being Element of Prop Q holds
( [p,q] in b2 iff p <==> q ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines PropRel QMAX_1:def 12 :
for Q being Quantum_Mechanics
for b2 being Equivalence_Relation of Prop Q holds
( b2 = PropRel Q iff for p, q being Element of Prop Q holds
( [p,q] in b2 iff p <==> q ) );

theorem :: QMAX_1:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th30: :: QMAX_1:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for B, C being Subset of (Prop Q) st B in Class (PropRel Q) & C in Class (PropRel Q) holds
for a, b, c, d being Element of Prop Q st a in B & b in B & c in C & d in C & a |- c holds
b |- d
proof end;

definition
let Q be Quantum_Mechanics;
func OrdRel Q -> Relation of Class (PropRel Q) means :Def13: :: QMAX_1:def 13
for B, C being Subset of (Prop Q) holds
( [B,C] in it iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) );
existence
ex b1 being Relation of Class (PropRel Q) st
for B, C being Subset of (Prop Q) holds
( [B,C] in b1 iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) )
proof end;
uniqueness
for b1, b2 being Relation of Class (PropRel Q) st ( for B, C being Subset of (Prop Q) holds
( [B,C] in b1 iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) ) ) & ( for B, C being Subset of (Prop Q) holds
( [B,C] in b2 iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def13 defines OrdRel QMAX_1:def 13 :
for Q being Quantum_Mechanics
for b2 being Relation of Class (PropRel Q) holds
( b2 = OrdRel Q iff for B, C being Subset of (Prop Q) holds
( [B,C] in b2 iff ( B in Class (PropRel Q) & C in Class (PropRel Q) & ( for p, q being Element of Prop Q st p in B & q in C holds
p |- q ) ) ) );

theorem :: QMAX_1:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th32: :: QMAX_1:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for p, q being Element of Prop Q holds
( p |- q iff [(Class (PropRel Q),p),(Class (PropRel Q),q)] in OrdRel Q )
proof end;

theorem Th33: :: QMAX_1:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for B, C being Subset of (Prop Q) st B in Class (PropRel Q) & C in Class (PropRel Q) holds
for p1, q1 being Element of Prop Q st p1 in B & q1 in B & 'not' p1 in C holds
'not' q1 in C
proof end;

theorem :: QMAX_1:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics
for B, C being Subset of (Prop Q) st B in Class (PropRel Q) & C in Class (PropRel Q) holds
for p, q being Element of Prop Q st 'not' p in C & 'not' q in C & p in B holds
q in B
proof end;

definition
let Q be Quantum_Mechanics;
func InvRel Q -> Function of Class (PropRel Q), Class (PropRel Q) means :Def14: :: QMAX_1:def 14
for p being Element of Prop Q holds it . (Class (PropRel Q),p) = Class (PropRel Q),('not' p);
existence
ex b1 being Function of Class (PropRel Q), Class (PropRel Q) st
for p being Element of Prop Q holds b1 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p)
proof end;
uniqueness
for b1, b2 being Function of Class (PropRel Q), Class (PropRel Q) st ( for p being Element of Prop Q holds b1 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p) ) & ( for p being Element of Prop Q holds b2 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def14 defines InvRel QMAX_1:def 14 :
for Q being Quantum_Mechanics
for b2 being Function of Class (PropRel Q), Class (PropRel Q) holds
( b2 = InvRel Q iff for p being Element of Prop Q holds b2 . (Class (PropRel Q),p) = Class (PropRel Q),('not' p) );

theorem :: QMAX_1:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: QMAX_1:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for Q being Quantum_Mechanics holds POI_Str(# (OrdRel Q),(InvRel Q) #) is_a_Quantuum_Logic_on Class (PropRel Q)
proof end;