defpred S₁[ set ] means ex W1, W2 being Subspace of V st
( W1 is Subspace of W2 & (dim W1) + 1 = k & dim W2 = k + 1 & $1 = pencil (W1,W2,k) );
set A = bool (k Subspaces_of V);
consider X being set such that
A1: for x being set holds
( x in X iff ( x in bool (k Subspaces_of V) & S₁[x] ) ) from XBOOLE_0:sch 1();
X c= bool (k Subspaces_of V)

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in X or a in bool (k Subspaces_of V) )
assume a in X ; :: thesis: a in bool (k Subspaces_of V)
hence a in bool (k Subspaces_of V) by A1; :: thesis: verum

end;

then reconsider X = X as Subset-Family of (k Subspaces_of V) ;
take X ; :: thesis: for X being set holds
( X in X iff ex W1, W2 being Subspace of V st
( W1 is Subspace of W2 & (dim W1) + 1 = k & dim W2 = k + 1 & X = pencil (W1,W2,k) ) )
let x be set ; :: thesis: ( x in X iff ex W1, W2 being Subspace of V st
( W1 is Subspace of W2 & (dim W1) + 1 = k & dim W2 = k + 1 & x = pencil (W1,W2,k) ) )
thus ( x in X implies ex W1, W2 being Subspace of V st
( W1 is Subspace of W2 & (dim W1) + 1 = k & dim W2 = k + 1 & x = pencil (W1,W2,k) ) ) by A1; :: thesis: ( ex W1, W2 being Subspace of V st
( W1 is Subspace of W2 & (dim W1) + 1 = k & dim W2 = k + 1 & x = pencil (W1,W2,k) ) implies x in X )
given W1, W2 being Subspace of V such that A2: ( W1 is Subspace of W2 & (dim W1) + 1 = k & dim W2 = k + 1 & x = pencil (W1,W2,k) ) ; :: thesis: x in X
thus x in X by A1, A2; :: thesis: verum