let R be Ring; :: thesis: for V being RightMod of R
for u, v being Vector of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being Vector of V st
( v1 in W & u - v1 = v )
let V be RightMod of R; :: thesis: for u, v being Vector of V
for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being Vector of V st
( v1 in W & u - v1 = v )
let u, v be Vector of V; :: thesis: for W being Submodule of V
for C being Coset of W st u in C & v in C holds
ex v1 being Vector of V st
( v1 in W & u - v1 = v )
let W be Submodule of V; :: thesis: for C being Coset of W st u in C & v in C holds
ex v1 being Vector of V st
( v1 in W & u - v1 = v )
let C be Coset of W; :: thesis: ( u in C & v in C implies ex v1 being Vector of V st
( v1 in W & u - v1 = v ) )
assume ( u in C & v in C ) ; :: thesis: ex v1 being Vector of V st
( v1 in W & u - v1 = v )
then ( C = u + W & C = v + W ) by Th74;
hence ex v1 being Vector of V st
( v1 in W & u - v1 = v ) by Th62; :: thesis: verum