let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V
for x being set holds
( x in W1 + W2 iff ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let V be RightMod of R; :: thesis: for W1, W2 being Submodule of V
for x being set holds
( x in W1 + W2 iff ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let W1, W2 be Submodule of V; :: thesis: for x being set holds
( x in W1 + W2 iff ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
let x be set ; :: thesis: ( x in W1 + W2 iff ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
thus ( x in W1 + W2 implies ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) ) :: thesis: ( ex v1, v2 being Vector of V 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 Vector of V 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 v, u is Vector of V : ( v in W1 & u in W2 ) } by Def1;
then consider v1, v2 being Vector of V such that
A1: ( x = v1 + v2 & v1 in W1 & v2 in W2 ) ;
take v1 ; :: thesis: ex v2 being Vector of V 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 Vector of V such that A2: ( v1 in W1 & v2 in W2 & x = v1 + v2 ) ; :: thesis: x in W1 + W2
x in { (v + u) where v, u is Vector of V : ( 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