let k be Element of NAT ; :: thesis: for X being non empty set st 0 < k & k + 1 c= card X holds
for T being Subset of st T is STAR holds
for S being Subset of st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } )
let X be non empty set ; :: thesis: ( 0 < k & k + 1 c= card X implies for T being Subset of st T is STAR holds
for S being Subset of st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } ) )
assume A1: ( 0 < k & k + 1 c= card X ) ; :: thesis: for T being Subset of st T is STAR holds
for S being Subset of st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } )
let T be Subset of ; :: thesis: ( T is STAR implies for S being Subset of st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } ) )
assume T is STAR ; :: thesis: for S being Subset of st S = meet T holds
( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } )
then consider S1 being Subset of such that
A2: ( card S1 = k - 1 & T = { A where A is Subset of : ( card A = k & S1 c= A ) } ) by Def4;
let S be Subset of ; :: thesis: ( S = meet T implies ( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } ) )
assume A3: S = meet T ; :: thesis: ( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } )
S1 = meet T by A1, A2, Th26;
hence ( card S = k - 1 & T = { A where A is Subset of : ( card A = k & S c= A ) } ) by A3, A2; :: thesis: verum