
theorem Th1:
  for X,Y,Z be RealLinearSpace for f be LinearOperator of X,Y
  for g be LinearOperator of Y,Z holds g*f is LinearOperator of X,Z
proof
  let X,Y,Z be RealLinearSpace;
  let f be LinearOperator of X,Y;
  let g be LinearOperator of Y,Z;
A1: now
    let v be VECTOR of X, a be Real;
    thus (g*f).(a*v) =g.(f.(a*v)) by FUNCT_2:15
      .=g.(a*f.v) by LOPBAN_1:def 5
      .=a*g.(f.v) by LOPBAN_1:def 5
      .=a*(g*f).(v) by FUNCT_2:15;
  end;
  now
    let v,w be VECTOR of X;
    thus (g*f).(v+w) =g.(f.(v+w)) by FUNCT_2:15
      .=g.(f.v+f.w) by VECTSP_1:def 20
      .=g.(f.v)+g.(f.w) by VECTSP_1:def 20
      .=(g*f).(v)+g.(f.w) by FUNCT_2:15
      .=(g*f).(v)+(g*f).(w) by FUNCT_2:15;
  end;
  hence thesis by A1,LOPBAN_1:def 5,VECTSP_1:def 20;
end;
