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 ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) 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 ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) 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 ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) 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 ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) 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 ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,(RealPoset [.0,1.]))
let a be Element of (RealPoset [.0,1.]); :: thesis: ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,(RealPoset [.0,1.]))
A1: { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } c= the carrier of (RealPoset [.0,1.])
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 (RealPoset [.0,1.]) )
assume d in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } ; :: thesis: d in the carrier of (RealPoset [.0,1.])
then ex t being Element of Y st d = (R . (x,t)) "/\" (S . (t,z)) ;
hence d in the carrier of (RealPoset [.0,1.]) ; :: thesis: verum
end;
set A = { (b "/\" a) where b is Element of (RealPoset [.0,1.]) : 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 (RealPoset [.0,1.]) : 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 (RealPoset [.0,1.]) : 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 (RealPoset [.0,1.]) : 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 (RealPoset [.0,1.]) : 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 (RealPoset [.0,1.]) 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 (RealPoset [.0,1.]) : 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 (RealPoset [.0,1.]) : 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 } ,(RealPoset [.0,1.]))) "/\" a by Th22
.= "\/" ( { (b "/\" a) where b is Element of (RealPoset [.0,1.]) : b in { ((R . (x,y)) "/\" (S . (y,z))) where y is Element of Y : verum } } ,(RealPoset [.0,1.])) by A1, Th15 ;
hence ((R (#) S) . (x,z)) "/\" a = "\/" ( { (((R . (x,y)) "/\" (S . (y,z))) "/\" a) where y is Element of Y : verum } ,(RealPoset [.0,1.])) by A2, TARSKI:2; :: thesis: verum