let E be RealLinearSpace; :: thesis: for F being binary-image-family of E
for B being non empty binary-image of E holds (meet F) (-) B = meet { (X (-) B) where X is binary-image of E : X in F }
let F be binary-image-family of E; :: thesis: for B being non empty binary-image of E holds (meet F) (-) B = meet { (X (-) B) where X is binary-image of E : X in F }
let B be non empty binary-image of E; :: thesis: (meet F) (-) B = meet { (X (-) B) where X is binary-image of E : X in F }
per cases ( F = {} or F <> {} ) ;
suppose A1: F = {} ; :: thesis: (meet F) (-) B = meet { (X (-) B) where X is binary-image of E : X in F }
reconsider Z = meet F as Subset of E ;
A2: meet F = {} by A1, SETFAM_1:def 1;
{ (X (-) B) where X is binary-image of E : X in F } = {}
proof
assume { (X (-) B) where X is binary-image of E : X in F } <> {} ; :: thesis: contradiction
then consider x being object such that
A3: x in { (X (-) B) where X is binary-image of E : X in F } by XBOOLE_0:def 1;
ex X being binary-image of E st
( x = X (-) B & X in F ) by A3;
hence contradiction by A1; :: thesis: verum
end;
then {} = meet { (X (-) B) where X is binary-image of E : X in F } by SETFAM_1:def 1;
hence (meet F) (-) B = meet { (X (-) B) where X is binary-image of E : X in F } by A2, Th2; :: thesis: verum
end;
suppose A4: F <> {} ; :: thesis: (meet F) (-) B = meet { (X (-) B) where X is binary-image of E : X in F }
then consider W0 being object such that
A5: W0 in F by XBOOLE_0:def 1;
reconsider W0 = W0 as binary-image of E by A5;
A6: W0 (-) B in { (W (-) B) where W is binary-image of E : W in F } by A5;
for x being object holds
( x in (meet F) (-) B iff x in meet { (W (-) B) where W is binary-image of E : W in F } )
proof
let x be object ; :: thesis: ( x in (meet F) (-) B iff x in meet { (W (-) B) where W is binary-image of E : W in F } )
hereby :: thesis: ( x in meet { (W (-) B) where W is binary-image of E : W in F } implies x in (meet F) (-) B )
assume x in (meet F) (-) B ; :: thesis: x in meet { (W (-) B) where W is binary-image of E : W in F }
then consider z being Element of E such that
A7: ( x = z & ( for b being Element of E st b in B holds
z - b in meet F ) ) ;
now :: thesis: for Y being set st Y in { (X (-) B) where X is binary-image of E : X in F } holds
x in Y
let Y be set ; :: thesis: ( Y in { (X (-) B) where X is binary-image of E : X in F } implies x in Y )
assume Y in { (X (-) B) where X is binary-image of E : X in F } ; :: thesis: x in Y
then consider X being binary-image of E such that
A8: ( Y = X (-) B & X in F ) ;
now :: thesis: for b being Element of E st b in B holds
z - b in X
let b be Element of E; :: thesis: ( b in B implies z - b in X )
assume b in B ; :: thesis: z - b in X
then A9: z - b in meet F by A7;
meet F c= X by A8, SETFAM_1:3;
hence z - b in X by A9; :: thesis: verum
end;
hence x in Y by A8, A7; :: thesis: verum
end;
hence x in meet { (W (-) B) where W is binary-image of E : W in F } by A6, SETFAM_1:def 1; :: thesis: verum
end;
assume A10: x in meet { (W (-) B) where W is binary-image of E : W in F } ; :: thesis: x in (meet F) (-) B
A11: for W being binary-image of E st W in F holds
x in W (-) B
proof
let W be binary-image of E; :: thesis: ( W in F implies x in W (-) B )
assume W in F ; :: thesis: x in W (-) B
then W (-) B in { (D (-) B) where D is binary-image of E : D in F } ;
hence x in W (-) B by A10, SETFAM_1:def 1; :: thesis: verum
end;
x in W0 (-) B by A5, A11;
then reconsider z = x as Element of E ;
for b being Element of E st b in B holds
z - b in meet F
proof
assume ex b being Element of E st
( b in B & not z - b in meet F ) ; :: thesis: contradiction
then consider b being Element of E such that
A12: ( b in B & not z - b in meet F ) ;
consider X being set such that
A13: ( X in F & not z - b in X ) by A4, A12, SETFAM_1:def 1;
reconsider X = X as binary-image of E by A13;
z in X (-) B by A13, A11;
then consider zz being Element of E such that
A14: ( z = zz & ( for b being Element of E st b in B holds
zz - b in X ) ) ;
thus contradiction by A14, A12, A13; :: thesis: verum
end;
hence x in (meet F) (-) B ; :: thesis: verum
end;
hence (meet F) (-) B = meet { (X (-) B) where X is binary-image of E : X in F } by TARSKI:2; :: thesis: verum
end;
end;