reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;
reserve B,C for Coset of W;

theorem Th56:
  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,Th55
    .= v2 + W by A2,Th55;
end;
