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 Th63:
  (ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W
proof
  thus (ex v st v1 in v + W & v2 in v + W) implies v1 - v2 in W
  proof
    given v such that
A1: v1 in v + W and
A2: v2 in v + W;
    consider u2 such that
A3: u2 in W and
A4: v2 = v + u2 by A2,Th42;
    consider u1 such that
A5: u1 in W and
A6: v1 = v + u1 by A1,Th42;
    v1 - v2 = (u1 + v) + ((- v) - u2) by A6,A4,VECTSP_1:17
      .= ((u1 + v) + (- v)) - u2 by RLVECT_1:def 3
      .= (u1 + (v + (- v))) - u2 by RLVECT_1:def 3
      .= (u1 + 0.V) - u2 by RLVECT_1:5
      .= u1 - u2 by RLVECT_1:4;
    hence thesis by A5,A3,Th23;
  end;
  assume v1 - v2 in W;
  then
A7: - (v1 - v2) in W by Th22;
  take v1;
  thus v1 in v1 + W by Th44;
  v1 + (- (v1 - v2)) = v1 + ((- v1) + v2) by VECTSP_1:17
    .= (v1 + (- v1)) + v2 by RLVECT_1:def 3
    .= 0.V + v2 by RLVECT_1:5
    .= v2 by RLVECT_1:4;
  hence thesis by A7;
end;
