:: YELLOW17 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

theorem Th1: :: YELLOW17:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for F being Function
for i, xi being set
for Ai being Subset of (F . i) st (proj F,i) " {xi} meets (proj F,i) " Ai holds
xi in Ai
proof end;

theorem Th2: :: YELLOW17:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for F, f being Function
for i, xi being set st xi in F . i & f in product F holds
f +* i,xi in product F
proof end;

theorem Th3: :: YELLOW17:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for F being Function
for i being set st i in dom F & product F <> {} holds
rng (proj F,i) = F . i
proof end;

theorem Th4: :: YELLOW17:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for F being Function
for i being set st i in dom F holds
(proj F,i) " (F . i) = product F
proof end;

theorem Th5: :: YELLOW17:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for F, f being Function
for i, xi being set st xi in F . i & i in dom F & f in product F holds
f +* i,xi in (proj F,i) " {xi}
proof end;

Lm1: for F, g being Function
for i1, i2, xi1 being set
for Ai2 being Subset of (F . i2) st i1 <> i2 & g in product F & g +* i1,xi1 in (proj F,i2) " Ai2 holds
g in (proj F,i2) " Ai2
proof end;

theorem Th6: :: YELLOW17:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for F, f being Function
for i1, i2, xi1 being set
for Ai2 being Subset of (F . i2) st xi1 in F . i1 & i1 in dom F & f in product F & i1 <> i2 holds
( f in (proj F,i2) " Ai2 iff f +* i1,xi1 in (proj F,i2) " Ai2 )
proof end;

theorem Th7: :: YELLOW17:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for F being Function
for i1, i2, xi1 being set
for Ai2 being Subset of (F . i2) st product F <> {} & xi1 in F . i1 & i1 in dom F & i2 in dom F & Ai2 <> F . i2 holds
( (proj F,i1) " {xi1} c= (proj F,i2) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) )
proof end;

scheme :: YELLOW17:sch 1
ElProductEx{ F1() -> non empty set , F2() -> non-Empty TopSpace-yielding ManySortedSet of F1(), P1[ set , set ] } :
ex f being Element of (product F2()) st
for i being Element of F1() holds P1[f . i,i]
provided
A1: for i being Element of F1() ex x being Element of (F2() . i) st P1[x,i]
proof end;

theorem Th8: :: YELLOW17:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for f being Element of (product J) holds (proj J,i) . f = f . i
proof end;

theorem Th9: :: YELLOW17:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for xi being Element of (J . i)
for Ai being Subset of (J . i) st (proj J,i) " {xi} meets (proj J,i) " Ai holds
xi in Ai
proof end;

theorem Th10: :: YELLOW17:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I holds (proj J,i) " ([#] (J . i)) = [#] (product J)
proof end;

theorem Th11: :: YELLOW17:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for xi being Element of (J . i)
for f being Element of (product J) holds f +* i,xi in (proj J,i) " {xi}
proof end;

theorem Th12: :: YELLOW17:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i1, i2 being Element of I
for xi1 being Element of (J . i1)
for Ai2 being Subset of (J . i2) st Ai2 <> [#] (J . i2) holds
( (proj J,i1) " {xi1} c= (proj J,i2) " Ai2 iff ( i1 = i2 & xi1 in Ai2 ) )
proof end;

theorem Th13: :: YELLOW17:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i1, i2 being Element of I
for xi1 being Element of (J . i1)
for Ai2 being Subset of (J . i2)
for f being Element of (product J) st i1 <> i2 holds
( f in (proj J,i2) " Ai2 iff f +* i1,xi1 in (proj J,i2) " Ai2 )
proof end;

theorem :: YELLOW17:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th15: :: YELLOW17:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct holds
( T is compact iff for F being Subset-Family of T st F is open & [#] T c= union F holds
ex G being Subset-Family of T st
( G c= F & [#] T c= union G & G is finite ) )
proof end;

theorem Th16: :: YELLOW17:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace
for B being prebasis of T holds
( T is compact iff for F being Subset of B st [#] T c= union F holds
ex G being finite Subset of F st [#] T c= union G )
proof end;

theorem Th17: :: YELLOW17:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for A being set st A in product_prebasis J holds
ex i being Element of I ex Ai being Subset of (J . i) st
( Ai is open & (proj J,i) " Ai = A )
proof end;

theorem Th18: :: YELLOW17:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for xi being Element of (J . i)
for A being set st A in product_prebasis J & (proj J,i) " {xi} c= A & not A = [#] (product J) holds
ex Ai being Subset of (J . i) st
( Ai <> [#] (J . i) & xi in Ai & Ai is open & A = (proj J,i) " Ai )
proof end;

theorem Th19: :: YELLOW17:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for Fi being non empty Subset-Family of (J . i) st [#] (J . i) c= union Fi holds
[#] (product J) c= union { ((proj J,i) " Ai) where Ai is Element of Fi : verum }
proof end;

theorem Th20: :: YELLOW17:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for xi being Element of (J . i)
for G being Subset of (product_prebasis J) st (proj J,i) " {xi} c= union G & ( for A being set st A in product_prebasis J & A in G holds
not (proj J,i) " {xi} c= A ) holds
[#] (product J) c= union G
proof end;

theorem Th21: :: YELLOW17:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for F being Subset of (product_prebasis J) st ( for G being finite Subset of F holds not [#] (product J) c= union G ) holds
for xi being Element of (J . i)
for G being finite Subset of F st (proj J,i) " {xi} c= union G holds
ex A being set st
( A in product_prebasis J & A in G & (proj J,i) " {xi} c= A )
proof end;

theorem Th22: :: YELLOW17:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for F being Subset of (product_prebasis J) st ( for G being finite Subset of F holds not [#] (product J) c= union G ) holds
for xi being Element of (J . i)
for G being finite Subset of F st (proj J,i) " {xi} c= union G holds
ex Ai being Subset of (J . i) st
( Ai <> [#] (J . i) & xi in Ai & (proj J,i) " Ai in G & Ai is open )
proof end;

theorem Th23: :: YELLOW17:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I
for i being Element of I
for F being Subset of (product_prebasis J) st ( for i being Element of I holds J . i is compact ) & ( for G being finite Subset of F holds not [#] (product J) c= union G ) holds
ex xi being Element of (J . i) st
for G being finite Subset of F holds not (proj J,i) " {xi} c= union G
proof end;

theorem :: YELLOW17:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for I being non empty set
for J being non-Empty TopSpace-yielding ManySortedSet of I st ( for i being Element of I holds J . i is compact ) holds
product J is compact
proof end;