let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for V1, V2 being Subset of V st V1 is linearly-closed & V2 is linearly-closed holds
V1 /\ V2 is linearly-closed
let V1, V2 be Subset of V; :: thesis: ( V1 is linearly-closed & V2 is linearly-closed implies V1 /\ V2 is linearly-closed )
assume that
A1: V1 is linearly-closed and
A2: V2 is linearly-closed ; :: thesis: V1 /\ V2 is linearly-closed
thus for v, u being Element of V st v in V1 /\ V2 & u in V1 /\ V2 holds
v + u in V1 /\ V2 :: according to VECTSP_4:def 1 :: thesis: for a being Element of GF
for v being Element of V st v in V1 /\ V2 holds
a * v in V1 /\ V2
proof
let v, u be Element of V; :: thesis: ( v in V1 /\ V2 & u in V1 /\ V2 implies v + u in V1 /\ V2 )
assume A3: ( v in V1 /\ V2 & u in V1 /\ V2 ) ; :: thesis: v + u in V1 /\ V2
then ( v in V2 & u in V2 ) by XBOOLE_0:def 4;
then A4: v + u in V2 by A2;
( v in V1 & u in V1 ) by A3, XBOOLE_0:def 4;
then v + u in V1 by A1;
hence v + u in V1 /\ V2 by A4, XBOOLE_0:def 4; :: thesis: verum
end;
let a be Element of GF; :: thesis: for v being Element of V st v in V1 /\ V2 holds
a * v in V1 /\ V2
let v be Element of V; :: thesis: ( v in V1 /\ V2 implies a * v in V1 /\ V2 )
assume A5: v in V1 /\ V2 ; :: thesis: a * v in V1 /\ V2
then v in V2 by XBOOLE_0:def 4;
then A6: a * v in V2 by A2;
v in V1 by A5, XBOOLE_0:def 4;
then a * v in V1 by A1;
hence a * v in V1 /\ V2 by A6, XBOOLE_0:def 4; :: thesis: verum