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 Th49:
  v in W implies (a * v) + W = the carrier of W
proof
  assume
A1: v in W;
  thus (a * v) + W c= the carrier of W
  proof
    let x be object;
    assume x in (a * v) + W;
    then consider u such that
A2: x = a * v + u and
A3: u in W;
    a * v in W by A1,Th21;
    then a * v + u in W by A3,Th20;
    hence thesis by A2;
  end;
  let x be object;
  assume
A4: x in the carrier of W;
  then
A5: x in W;
  the carrier of W c= the carrier of V by Def2;
  then reconsider y = x as Element of V by A4;
A6: a * v + (y - a * v) = (y + a * v) - a * v by RLVECT_1:def 3
    .= y + (a * v - a * v) by RLVECT_1:def 3
    .= y + 0.V by RLVECT_1:15
    .= x;
  a * v in W by A1,Th21;
  then y - a * v in W by A5,Th23;
  hence thesis by A6;
end;
