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