reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th86:
  for X,Y,Z be RealLinearSpace,
          L be LinearOperator of X,Y,
          K be LinearOperator of Y,Z
  holds K*L is LinearOperator of X,Z
  proof
    let X,Y,Z be RealLinearSpace,
            L be LinearOperator of X,Y,
            K be LinearOperator of Y,Z;
    reconsider T = K*L as Function of X,Z;

    for x, y be Element of X
    holds T . (x + y) = (T . x) + (T . y)
    proof
      let x, y be Element of X;
      A1: L is additive;
      A2: K is additive;

      thus T.(x + y)
       = K.(L.(x + y)) by FUNCT_2:15
      .= K.(L.x + L.y) by A1
      .= K.(L.x) + K.(L.y) by A2
      .= T.x + K.(L.y) by FUNCT_2:15
      .= T.x + T.y by FUNCT_2:15;
    end;
    then
    A3: T is additive;

    for a be Real,
        x be VECTOR of X
    holds T . (a * x) = a * (T . x)
    proof
      let a be Real;
      let x be VECTOR of X;
      thus T . (a * x)
       = K.(L.(a * x)) by FUNCT_2:15
      .= K.(a * L.x) by LOPBAN_1:def 5
      .= a * K.(L.x) by LOPBAN_1:def 5
      .= a * T.x by FUNCT_2:15;
    end;
    then
    T is homogeneous;
    hence K * L is LinearOperator of X,Z by A3;
  end;
