let k be Element of NAT ; :: thesis: for X being non empty set st k + 1 c= card X holds

for A being finite Subset of X st card A = k - 1 holds

^^ (A,X,k) is STAR

let X be non empty set ; :: thesis: ( k + 1 c= card X implies for A being finite Subset of X st card A = k - 1 holds

^^ (A,X,k) is STAR )

assume A1: k + 1 c= card X ; :: thesis: for A being finite Subset of X st card A = k - 1 holds

^^ (A,X,k) is STAR

let A be finite Subset of X; :: thesis: ( card A = k - 1 implies ^^ (A,X,k) is STAR )

assume A2: card A = k - 1 ; :: thesis: ^^ (A,X,k) is STAR

^^ (A,X,k) = ^^ (A,X) by A1, A2, Def13;

hence ^^ (A,X,k) is STAR by A2; :: thesis: verum

for A being finite Subset of X st card A = k - 1 holds

^^ (A,X,k) is STAR

let X be non empty set ; :: thesis: ( k + 1 c= card X implies for A being finite Subset of X st card A = k - 1 holds

^^ (A,X,k) is STAR )

assume A1: k + 1 c= card X ; :: thesis: for A being finite Subset of X st card A = k - 1 holds

^^ (A,X,k) is STAR

let A be finite Subset of X; :: thesis: ( card A = k - 1 implies ^^ (A,X,k) is STAR )

assume A2: card A = k - 1 ; :: thesis: ^^ (A,X,k) is STAR

^^ (A,X,k) = ^^ (A,X) by A1, A2, Def13;

hence ^^ (A,X,k) is STAR by A2; :: thesis: verum