reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;
reserve B,C for Coset of W;

theorem Th54:
  u in v1 + W & u in v2 + W implies v1 + W = v2 + W
proof
  assume that
A1: u in v1 + W and
A2: u in v2 + W;
  thus v1 + W = u + W by A1,Th53
    .= v2 + W by A2,Th53;
end;
