
theorem Th18:
  for X be RealLinearSpace
  holds RLSp2RVSp((0).X) = (0).RLSp2RVSp(X)
  proof
    let X be RealLinearSpace;
    set Y = (0).X;

    A1: the carrier of (0).RLSp2RVSp(X)
     = { 0.(RLSp2RVSp(X)) } by VECTSP_4:def 3
    .= { 0.X };

    A2: the addF of RLSp2RVSp(Y)
     = (the addF of RLSp2RVSp(X)) ||
        the carrier of RLSp2RVSp(Y) by VECTSP_4:def 2
    .= (the addF of RLSp2RVSp(X)) ||
        the carrier of (0).(RLSp2RVSp(X)) by A1,RLSUB_1:def 3
    .= the addF of (0).(RLSp2RVSp(X)) by VECTSP_4:def 2;

    A3: 0.(RLSp2RVSp(Y))
     = 0.(RLSp2RVSp(X)) by VECTSP_4:def 2
    .= 0.((0).RLSp2RVSp(X)) by VECTSP_4:def 2;

    MultF_Real* Y
     = (the Mult of X) | [:REAL, the carrier of (Y):] by RLSUB_1:def 2
    .= (the lmult of RLSp2RVSp(X)) |
      [:the carrier of F_Real, the carrier of (0).(RLSp2RVSp(X)):]
        by A1,RLSUB_1:def 3
    .= the lmult of (0).(RLSp2RVSp(X)) by VECTSP_4:def 2;
    hence thesis by A1,A2,A3,RLSUB_1:def 3;
  end;
