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 Th45:
  v + (0).V = {v}
proof
  thus v + (0).V c= {v}
  proof
    let x be object;
    assume x in v + (0).V;
    then consider u such that
A1: x = v + u and
A2: u in (0).V;
A3: the carrier of (0).V = {0.V} by Def3;
    u in the carrier of (0).V by A2;
    then u = 0.V by A3,TARSKI:def 1;
    then x = v by A1;
    hence thesis by TARSKI:def 1;
  end;
  let x be object;
  assume x in {v};
  then
A4: x = v by TARSKI:def 1;
  0.V in (0).V & v = v + 0.V by Th17;
  hence thesis by A4;
end;
