
theorem Th16:
  for X be RealLinearSpace
  for Y1,Y2 be Subspace of X
  holds RLSp2RVSp(Y1 + Y2) = RLSp2RVSp(Y1) + RLSp2RVSp(Y2)
  proof
    let X be RealLinearSpace;
    let Y1,Y2 be Subspace of X;
    set Y12 = Y1 + Y2;

    A1: the carrier of Y12
      = {(v + u) where u, v is VECTOR of X : (v in Y1 & u in Y2)}
      by RLSUB_2:def 1;

    A2: the carrier of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2))
     = { (v + u) where u, v is Element of RLSp2RVSp(X)
        : (v in RLSp2RVSp(Y1) & u in RLSp2RVSp(Y2)) } by VECTSP_5:def 1;

    A3: for x be object
        holds
          x in the carrier of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2))
            iff
          x in the carrier of RLSp2RVSp(Y12)
    proof
      let x be object;
      hereby
        assume x in the carrier of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2));
        then consider u, v be Element of RLSp2RVSp(X) such that
        A4: x = v + u
          & v in RLSp2RVSp(Y1)
          & u in RLSp2RVSp(Y2) by A2;

        reconsider u1 = u, v1 = v as Element of X;
        A5: v1 in Y1 & u1 in Y2 by A4;

        v + u = v1 + u1;
        hence x in the carrier of RLSp2RVSp(Y12) by A1,A4,A5;
      end;
      assume x in the carrier of RLSp2RVSp(Y12);
      then consider u, v be VECTOR of X such that
      A6: x = v + u & v in Y1 & u in Y2 by A1;
      reconsider v1 = v, u1 = u as Element of RLSp2RVSp(X);
      A7: v in RLSp2RVSp(Y1) & u in RLSp2RVSp(Y2) by A6;

      v + u = v1 + u1;
      hence x in the carrier of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2)) by A2,A6,A7;
    end;

    A8: the addF of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2))
     = (the addF of RLSp2RVSp (X)) ||
          the carrier of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2)) by VECTSP_4:def 2
    .= (the addF of RLSp2RVSp (X)) || the carrier of RLSp2RVSp(Y12)
          by A3,TARSKI:2
    .= the addF of RLSp2RVSp(Y1 + Y2) by VECTSP_4:def 2;

    A9: 0.(RLSp2RVSp(Y1) + RLSp2RVSp(Y2))
     = 0.(RLSp2RVSp(X)) by VECTSP_4:def 2
    .= 0.(RLSp2RVSp(Y1 + Y2)) by VECTSP_4:def 2;

    MultF_Real* (Y1 + Y2)
     = (the Mult of X) | [:REAL, the carrier of (Y1+Y2):] by RLSUB_1:def 2
    .= (the lmult of RLSp2RVSp(X)) |
        [:the carrier of F_Real,
          the carrier of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2)):] by A3,TARSKI:2
    .= the lmult of (RLSp2RVSp(Y1) + RLSp2RVSp(Y2)) by VECTSP_4:def 2;
    hence thesis by A3,A8,A9,TARSKI:2;
  end;
