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
for a being Element of (RealPoset [.0,1.]) holds a "/\" ((R (#) S) . (x,z)) = "\/" ( { ((a "/\" (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
for a being Element of (RealPoset [.0,1.]) holds a "/\" ((R (#) S) . (x,z)) = "\/" ( { ((a "/\" (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
for a being Element of (RealPoset [.0,1.]) holds a "/\" ((R (#) S) . (x,z)) = "\/" ( { ((a "/\" (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
for a being Element of (RealPoset [.0,1.]) holds a "/\" ((R (#) S) . (x,z)) = "\/" ( { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } ,(RealPoset [.0,1.]))
let z be Element of Z; :: thesis: for a being Element of (RealPoset [.0,1.]) holds a "/\" ((R (#) S) . (x,z)) = "\/" ( { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } ,(RealPoset [.0,1.]))
let a be Element of (RealPoset [.0,1.]); :: thesis: a "/\" ((R (#) S) . (x,z)) = "\/" ( { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } ,(RealPoset [.0,1.]))
set A = { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } ;
set B = { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ;
for b being object holds
( b in { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } iff b in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } )

proof

let b be object ; :: thesis: ( b in { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } iff b in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } )

hereby :: thesis: ( b in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } implies b in { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } )

assume b in { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } ; :: thesis: b in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum }
then consider y being Element of Y such that
A1: b = (a "/\" (R . (x,y))) "/\" (S . (y,z)) ;
b = ((R . (x,y)) "/\" (S . (y,z))) "/\" a by A1, LATTICE3:16;
hence b in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ; :: thesis: verum

end;

assume b in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ; :: thesis: b in { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum }
then consider y being Element of Y such that
A2: b = ((R . (x,y)) "/\" (S . (y,z))) "/\" a ;
b = (a "/\" (R . (x,y))) "/\" (S . (y,z)) by A2, LATTICE3:16;
hence b in { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } ; :: thesis: verum

end;

then { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } = { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } by TARSKI:2;
hence a "/\" ((R (#) S) . (x,z)) = "\/" ( { ((a "/\" (R . (x,y))) "/\" (S . (y,z))) where y is Element of Y : verum } ,(RealPoset [.0,1.])) by Lm5; :: thesis: verum