reserve X, Y for RealNormSpace;

theorem Th8:
  for V be Subset of X,x be Point of X, V1 be Subset of
LinearTopSpaceNorm X, x1 be Point of LinearTopSpaceNorm X st V=V1 & x=x1 holds
  x+V=x1+V1
proof
  let V be Subset of X,x be Point of X, V1 be Subset of LinearTopSpaceNorm X,
  x1 be Point of LinearTopSpaceNorm X such that
A1: V=V1 and
A2: x=x1;
  thus x+V c= x1+V1
  proof
    let z be object;
    assume z in x+V;
    then consider u be Point of X such that
A3: z=x+u and
A4: u in V;
    reconsider u1=u as Point of LinearTopSpaceNorm X by A1,A4;
    x+u=x1+u1 by A2,NORMSP_2:def 4;
    hence thesis by A1,A3,A4;
  end;
  let z be object;
  assume z in x1+V1;
  then consider u1 be Point of LinearTopSpaceNorm X such that
A5: z=x1+u1 and
A6: u1 in V1;
  reconsider u=u1 as Point of X by A1,A6;
  x1+u1=x+u by A2,NORMSP_2:def 4;
  hence thesis by A1,A5,A6;
end;
