let V be RealUnitarySpace; :: thesis: for W being Subspace of V
for C being Coset of W
for u, v being VECTOR of V st u in C & v in C holds
ex v1 being VECTOR of V st
( v1 in W & u + v1 = v )
let W be Subspace of V; :: thesis: for C being Coset of W
for u, v being VECTOR of V 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: for u, v being VECTOR of V 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: ( 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 Th72;
hence ex v1 being VECTOR of V st
( v1 in W & u + v1 = v ) by Th60; :: thesis: verum