:: FUZZY_3 semantic presentation

registration
let C be non empty set ;
cluster -> quasi_total Membership_Func of a1;
coherence
for b1 being Membership_Func of C holds b1 is quasi_total
proof end;
end;

definition
let C1 be non empty set , C2 be non empty set ;
mode RMembership_Func of a1,a2 is Membership_Func of [:a1,a2:];
end;

definition
let C1 be non empty set , C2 be non empty set ;
func Zmf c1,c2 -> RMembership_Func of a1,a2 equals :: FUZZY_3:def 1
chi {} ,[:C1,C2:];
correctness
coherence
chi {} ,[:C1,C2:] is RMembership_Func of C1,C2
;
by FUZZY_1:13;
func Umf c1,c2 -> RMembership_Func of a1,a2 equals :: FUZZY_3:def 2
chi [:C1,C2:],[:C1,C2:];
correctness
coherence
chi [:C1,C2:],[:C1,C2:] is RMembership_Func of C1,C2
;
by FUZZY_1:2;
end;

:: deftheorem Def1 defines Zmf FUZZY_3:def 1 :
for C1, C2 being non empty set holds Zmf C1,C2 = chi {} ,[:C1,C2:];

:: deftheorem Def2 defines Umf FUZZY_3:def 2 :
for C1, C2 being non empty set holds Umf C1,C2 = chi [:C1,C2:],[:C1,C2:];

theorem Th1: :: FUZZY_3:1
for C1, C2 being non empty set holds Zmf C1,C2 = EMF [:C1,C2:] by FUZZY_1:def 7;

theorem Th2: :: FUZZY_3:2
for C1, C2 being non empty set holds Umf C1,C2 = UMF [:C1,C2:] by FUZZY_1:def 8;

theorem Th3: :: FUZZY_3:3
for C1, C2 being non empty set
for x being Element of [:C1,C2:]
for h being RMembership_Func of C1,C2 holds
( (Zmf C1,C2) . x <= h . x & h . x <= (Umf C1,C2) . x )
proof end;

theorem Th4: :: FUZZY_3:4
for C1, C2 being non empty set
for f being RMembership_Func of C1,C2 holds
( max f,(Umf C1,C2) = Umf C1,C2 & min f,(Umf C1,C2) = f & max f,(Zmf C1,C2) = f & min f,(Zmf C1,C2) = Zmf C1,C2 )
proof end;

theorem Th5: :: FUZZY_3:5
for C1, C2 being non empty set holds
( 1_minus (Zmf C1,C2) = Umf C1,C2 & 1_minus (Umf C1,C2) = Zmf C1,C2 )
proof end;

theorem Th6: :: FUZZY_3:6
for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2 st f \ g = Zmf C1,C2 holds
g c=
proof end;

theorem Th7: :: FUZZY_3:7
for C1, C2 being non empty set
for f, g being RMembership_Func of C1,C2 st min f,g = Zmf C1,C2 holds
f \ g = f
proof end;