let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W1, W2 being Subspace of M
for x being object holds
( x in W1 + W2 iff ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for W1, W2 being Subspace of M
for x being object holds
( x in W1 + W2 iff ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let W1, W2 be Subspace of M; :: thesis: for x being object holds
( x in W1 + W2 iff ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let x be object ; :: thesis: ( x in W1 + W2 iff ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
thus ( x in W1 + W2 implies ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) ) :: thesis: ( ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) implies x in W1 + W2 )

proof

assume x in W1 + W2 ; :: thesis: ex v1, v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 )
then x in the carrier of (W1 + W2) by STRUCT_0:def 5;
then x in { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } by Def1;
then consider v2, v1 being Element of M such that
A1: ( x = v1 + v2 & v1 in W1 & v2 in W2 ) ;
take v1 ; :: thesis: ex v2 being Element of M st
( v1 in W1 & v2 in W2 & x = v1 + v2 )
take v2 ; :: thesis: ( v1 in W1 & v2 in W2 & x = v1 + v2 )
thus ( v1 in W1 & v2 in W2 & x = v1 + v2 ) by A1; :: thesis: verum

end;

given v1, v2 being Element of M such that A2: ( v1 in W1 & v2 in W2 & x = v1 + v2 ) ; :: thesis: x in W1 + W2
x in { (v + u) where u, v is Element of M : ( v in W1 & u in W2 ) } by A2;
then x in the carrier of (W1 + W2) by Def1;
hence x in W1 + W2 by STRUCT_0:def 5; :: thesis: verum