set S = { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } ;
take { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } ; :: thesis: for x being object holds
( x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict free Submodule of V st
( W = x & rank W = n ) )

for x being object holds
( x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict free Submodule of V st
( W = x & rank W = n ) )
proof
let x be object ; :: thesis: ( x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict free Submodule of V st
( W = x & rank W = n ) )

hereby :: thesis: ( ex W being strict free Submodule of V st
( W = x & rank W = n ) implies x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } )
assume x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } ; :: thesis: ex W being strict free Submodule of V st
( W = x & rank W = n )

then A1: ex A being finite Subset of V st
( x = Lin A & A is linearly-independent & card A = n ) ;
then reconsider W = x as strict free Submodule of V ;
rank W = n by ;
hence ex W being strict free Submodule of V st
( W = x & rank W = n ) ; :: thesis: verum
end;
given W being strict free Submodule of V such that A2: W = x and
A3: rank W = n ; :: thesis: x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) }
consider A being finite Subset of W such that
A4: A is Basis of W by ZMODUL03:def 3;
reconsider A = A as Subset of W ;
reconsider B = A as linearly-independent Subset of V by ;
A5: x = Lin A by
.= Lin B by ZMODUL03:20 ;
then card B = n by ;
hence x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } by A5; :: thesis: verum
end;
hence for x being object holds
( x in { (Lin A) where A is finite Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict free Submodule of V st
( W = x & rank W = n ) ) ; :: thesis: verum