 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem
  for R being Ring
  for X, Y be LeftMod of R, l be Linear_Combination of X,
  L be linear-transformation of X, Y st L is bijective
  holds L@*l = l*L"
  proof
    let R be Ring;
    let X, Y be LeftMod of R,
    l be Linear_Combination of X,
    L be linear-transformation of X, Y;
    assume
    A1: L is bijective; then
    Q4: rng L = the carrier of Y by FUNCT_2:def 3;
    then reconsider K = L" as Function of Y, X by A1,FUNCT_2:25;
    P3: dom (l*K) = the carrier of Y by FUNCT_2:def 1;
    X1: L|(Carrier l) is one-to-one by A1,FUNCT_1:52;
    then P4: L .: (Carrier l) = Carrier ( L@*l) by ZMODUL05:56;
    for a being Element of Y holds (L@*l).a = (l*K).a
    proof
      let a be Element of Y;
      per cases;
      suppose X0: not a in Carrier (L@*l);
        reconsider v = K.a as Element of X;
        X4: L.v = a by Q4,A1,FUNCT_1:35;
        Y2: not v in Carrier l by X0,X4,FUNCT_2:35,P4;
        (l*K).a = l.v by P3,FUNCT_1:12
        .= 0.R by Y2;
        hence thesis by X0;
      end;
      suppose a in Carrier (L@*l);
        then consider v being object such that
        X5: v in dom L & v in (Carrier l) & a = L.v by FUNCT_1:def 6,P4;
        reconsider v as Element of X by X5;
        X6: K.a = v by A1,FUNCT_1:34,X5;
        thus (L@*l).a = l.v by ZMODUL05:54,X1,X5
        .= (l*K).a by P3,FUNCT_1:12,X6;
      end;
    end;
    hence thesis by FUNCT_2:def 8;
  end;
