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

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

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 a be Element of GF; :: thesis: for v being Element of V st v in V1 /\ V2 holds
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;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

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