begin
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
Lm1:
for X, x being set st x in X holds
(X \ {x}) \/ {x} = X
theorem Th5:
theorem Th6:
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
theorem Th13:
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
begin
theorem Th22:
theorem Th23:
begin
theorem Th24:
theorem
theorem Th26:
theorem Th27:
theorem Th28:
begin
:: deftheorem VECTSP_9:def 1 :
canceled;
:: deftheorem Def2 defines dim VECTSP_9:def 2 :
for GF being Field
for V being VectSp of GF st V is finite-dimensional holds
for b3 being Nat holds
( b3 = dim V iff for I being Basis of V holds b3 = card I );
theorem Th29:
theorem Th30:
theorem Th31:
theorem
theorem Th33:
theorem
theorem
theorem Th36:
theorem
theorem
Lm2:
for GF being Field
for n being Nat
for V being finite-dimensional VectSp of GF st n <= dim V holds
ex W being strict Subspace of V st dim W = n
theorem
:: deftheorem Def3 defines Subspaces_of VECTSP_9:def 3 :
for GF being Field
for V being finite-dimensional VectSp of GF
for n being Nat
for b4 being set holds
( b4 = n Subspaces_of V iff for x being set holds
( x in b4 iff ex W being strict Subspace of V st
( W = x & dim W = n ) ) );
theorem
theorem
theorem