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 Th42:
  x in v + W iff ex u st u in W & x = v + u
proof
  thus x in v + W implies ex u st u in W & x = v + u
  proof
    assume x in v + W;
    then consider u such that
A1: x = v + u & u in W;
    take u;
    thus thesis by A1;
  end;
  given u such that
A2: u in W & x = v + u;
  thus thesis by A2;
end;
