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
  u in v + W iff ex v1 st v1 in W & u = v - v1
proof
  thus u in v + W implies ex v1 st v1 in W & u = v - v1
  proof
    assume u in v + W;
    then consider v1 such that
A1: u = v + v1 and
A2: v1 in W;
    take x = - v1;
    thus x in W by A2,Th22;
    thus thesis by A1;
  end;
  given v1 such that
A3: v1 in W and
A4: u = v - v1;
  - v1 in W by A3,Th22;
  hence thesis by A4;
end;
