let R be Ring; :: thesis: for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #) iff for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
let V be RightMod of R; :: thesis: for W1, W2 being Submodule of V holds
( W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #) iff for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
let W1, W2 be Submodule of V; :: thesis: ( W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #) iff for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
thus ( W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #) implies for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) :: thesis: ( ( for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) implies W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #) )

proof

assume A1: W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #) ; :: thesis: for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
let v be Vector of V; :: thesis: ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 )
v in (Omega). V by RLVECT_1:3;
hence ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) by A1, Th5; :: thesis: verum

end;

assume A2: for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ; :: thesis: W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #)

now

thus W1 + W2 is Submodule of (Omega). V by Lm5; :: thesis: for u being Vector of V holds u in W1 + W2
let u be Vector of V; :: thesis: u in W1 + W2
ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & u = v1 + v2 ) by A2;
hence u in W1 + W2 by Th5; :: thesis: verum

end;

hence W1 + W2 = RightModStr(# the carrier of V,the U7 of V,the U2 of V,the rmult of V #) by RMOD_2:40; :: thesis: verum