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 () holds ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,())

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 () holds ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,())

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 () holds ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,())

let x be Element of X; :: thesis: for z being Element of Z
for a being Element of () holds ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,())

let z be Element of Z; :: thesis: for a being Element of () holds ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,())
let a be Element of (); :: thesis: ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,())
A1: { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } c= the carrier of ()
proof
let d be object ; :: according to TARSKI:def 3 :: thesis: ( not d in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } or d in the carrier of () )
assume d in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } ; :: thesis: d in the carrier of ()
then ex t being Element of Y st d = (R . (x,t)) "/\" (S . (t,z)) ;
hence d in the carrier of () ; :: thesis: verum
end;
set A = { (b "/\" a) where b is Element of () : b in { ((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 } ;
A2: for c being object holds
( c in { (b "/\" a) where b is Element of () : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } } iff c in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } )
proof
let c be object ; :: thesis: ( c in { (b "/\" a) where b is Element of () : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } } iff c in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } )
hereby :: thesis: ( c in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } implies c in { (b "/\" a) where b is Element of () : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } } )
assume c in { (b "/\" a) where b is Element of () : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } } ; :: thesis: c in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum }
then consider b being Element of () such that
A3: c = b "/\" a and
A4: b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } ;
ex y being Element of Y st b = (R . (x,y)) "/\" (S . (y,z)) by A4;
hence c in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } by A3; :: thesis: verum
end;
assume c in { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ; :: thesis: c in { (b "/\" a) where b is Element of () : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } }
then consider y being Element of Y such that
A5: c = ((R . (x,y)) "/\" (S . (y,z))) "/\" a ;
(R . (x,y)) "/\" (S . (y,z)) in { ((R . (x,y1)) "/\" (S . (y1,z))) where y1 is Element of Y : verum } ;
hence c in { (b "/\" a) where b is Element of () : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } } by A5; :: thesis: verum
end;
((R (#) S) . (x,z)) "/\" a = ("\/" ( { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } ,())) "/\" a by Th22
.= "\/" ( { (b "/\" a) where b is Element of () : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } } ,()) by ;
hence ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,()) by ; :: thesis: verum