reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:REAL,D:],D;
reserve B,C for Coset of W;

theorem Th55:
  v + W = (- v) + W iff v in W
proof
  thus v + W = (- v) + W implies v in W
  proof
    assume v + W = (- v) + W;
    then v in (- v) + W by Th43;
    then consider u such that
A1: v = - v + u and
A2: u in W;
  reconsider dwa=2 as Real;
    0.V = v - (- v + u) by A1,RLVECT_1:15
      .= (v - (- v)) - u by RLVECT_1:27
      .= (v + v) - u
      .= (1 * v + v) - u by RLVECT_1:def 8
      .= (1 * v + 1 * v) - u by RLVECT_1:def 8
      .= ((1 + 1) * v) - u by RLVECT_1:def 6
      .= 2 * v - u;
    then 2" * (2 * v) = 2" * u by RLVECT_1:21;
    then (2" * 2) * v = 2" * u by RLVECT_1:def 7;
    then v = dwa" * u by RLVECT_1:def 8;
    hence thesis by A2,Th21;
  end;
  assume
A3: v in W;
  then v + W = the carrier of W by Lm3;
  hence thesis by A3,Th51;
end;
