let X, Y, Z be non empty set ; :: thesis: for R being RMembership_Func of X,Y
for S being RMembership_Func of Y,Z
for x being Element of X
for z being Element of Z holds (R (#) S) . x,z = "\/" { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ,(RealPoset [.0 ,1.])
let R be RMembership_Func of X,Y; :: thesis: for S being RMembership_Func of Y,Z
for x being Element of X
for z being Element of Z holds (R (#) S) . x,z = "\/" { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ,(RealPoset [.0 ,1.])
let S be RMembership_Func of Y,Z; :: thesis: for x being Element of X
for z being Element of Z holds (R (#) S) . x,z = "\/" { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ,(RealPoset [.0 ,1.])
let x be Element of X; :: thesis: for z being Element of Z holds (R (#) S) . x,z = "\/" { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ,(RealPoset [.0 ,1.])
let z be Element of Z; :: thesis: (R (#) S) . x,z = "\/" { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ,(RealPoset [.0 ,1.])
set L = { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ;
[x,z] in [:X,Z:] ;
then A1: (R (#) S) . x,z = upper_bound (rng (min R,S,x,z)) by FUZZY_4:def 3;
A2: for b being Element of (RealPoset [.0 ,1.]) st b is_>=_than { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } holds
(R (#) S) . x,z <<= b

proof

let b be Element of (RealPoset [.0 ,1.]); :: thesis: ( b is_>=_than { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } implies (R (#) S) . x,z <<= b )
assume A3: b is_>=_than { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ; :: thesis: (R (#) S) . x,z <<= b
A4: rng (min R,S,x,z) c= [.0 ,1.] by RELAT_1:def 19;
A5: { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } = rng (min R,S,x,z) by Lm5;
A6: for r being real number st r in rng (min R,S,x,z) holds
r <= b

proof

let r be real number ; :: thesis: ( r in rng (min R,S,x,z) implies r <= b )
assume A7: r in rng (min R,S,x,z) ; :: thesis: r <= b
then reconsider r = r as Element of (RealPoset [.0 ,1.]) by A4, Def3;
r <<= b by A3, A5, A7, LATTICE3:def 9;
hence r <= b by Th3; :: thesis: verum

end;

rng (min R,S,x,z) <> {} by Lm5;
then upper_bound (rng (min R,S,x,z)) <= b by A6, TOPMETR3:1;
hence (R (#) S) . x,z <<= b by A1, Th3; :: thesis: verum

end;

for b being Element of (RealPoset [.0 ,1.]) st b in { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } holds
(R (#) S) . [x,z] >>= b

proof

let b be Element of (RealPoset [.0 ,1.]); :: thesis: ( b in { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } implies (R (#) S) . [x,z] >>= b )
assume b in { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ; :: thesis: (R (#) S) . [x,z] >>= b
then consider y being Element of Y such that
A8: b = (R . x,y) "/\" (S . y,z) and
verum ;
reconsider b = b as Real by Lm4;
( dom (min R,S,x,z) = Y & b = (min R,S,x,z) . y ) by A8, FUNCT_2:def 1, FUZZY_4:def 2;
then b <= upper_bound (rng (min R,S,x,z)) by FUZZY_4:1;
hence (R (#) S) . [x,z] >>= b by A1, Th3; :: thesis: verum

end;

then (R (#) S) . [x,z] is_>=_than { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } by LATTICE3:def 9;
hence (R (#) S) . x,z = "\/" { ((R . x,y) "/\" (S . y,z)) where y is Element of Y : verum } ,(RealPoset [.0 ,1.]) by A2, YELLOW_0:32; :: thesis: verum