set A = bool (m Subspaces_of V);
defpred S₁[ set ] means ex W being Subspace of V st
( dim W = n & $1 = m Subspaces_of W );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in bool (m Subspaces_of V) & S₁[x] ) ) from XBOOLE_0:sch 1();
X c= bool (m Subspaces_of V)

proof

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

end;

then reconsider X = X as Subset-Family of (m Subspaces_of V) ;
take X ; :: thesis: for X being set holds
( X in X iff ex W being Subspace of V st
( dim W = n & X = m Subspaces_of W ) )
let x be set ; :: thesis: ( x in X iff ex W being Subspace of V st
( dim W = n & x = m Subspaces_of W ) )
thus ( x in X implies ex W being Subspace of V st
( dim W = n & x = m Subspaces_of W ) ) by A1; :: thesis: ( ex W being Subspace of V st
( dim W = n & x = m Subspaces_of W ) implies x in X )
given W being Subspace of V such that A2: ( dim W = n & x = m Subspaces_of W ) ; :: thesis: x in X
x c= m Subspaces_of V by A2, VECTSP_9:42;
hence x in X by A1, A2; :: thesis: verum