
theorem
  for X be RealLinearSpace
  for Y1, Y2 be Subspace of X
    st Y1 /\ Y2 = (0).X
  holds
    for B1 be linearly-independent Subset of Y1
    for B2 be linearly-independent Subset of Y2
    holds B1 \/ B2 is linearly-independent Subset of (Y1 + Y2)
  proof
    let X be RealLinearSpace;
    let Y1, Y2 be Subspace of X;
    assume
    A1: Y1 /\ Y2 = (0).X;

    let B1 be linearly-independent Subset of Y1;
    let B2 be linearly-independent Subset of Y2;

    A2: RLSp2RVSp(Y1) /\ RLSp2RVSp(Y2)
     = RLSp2RVSp(Y1 /\ Y2) by Th17
    .= RLSp2RVSp((0).X) by A1
    .= (0).RLSp2RVSp(X) by Th18;

    A3: B1 is linearly-independent Subset of RLSp2RVSp(Y1)
      by REAL_NS2:78;
    A4: B2 is linearly-independent Subset of RLSp2RVSp(Y2)
      by REAL_NS2:78;

    RLSp2RVSp(Y1 + Y2) = RLSp2RVSp(Y1) + RLSp2RVSp(Y2) by Th16;

    then B1 \/ B2 is linearly-independent Subset of RLSp2RVSp(Y1 + Y2)
      by A2,A3,A4,MATRLIN2:2;

    hence B1 \/ B2 is linearly-independent Subset of (Y1 + Y2)
      by REAL_NS2:78;
  end;
