
theorem Th1:
  for X,Y,Z be ComplexLinearSpace, f be LinearOperator of X,Y, g be
  LinearOperator of Y,Z holds g*f is LinearOperator of X,Z
proof
  let X,Y,Z be ComplexLinearSpace;
  let f be LinearOperator of X,Y;
  let g be LinearOperator of Y,Z;
A1: now
    let v be VECTOR of X, z be Complex;
    thus (g*f).(z*v) =g.(f.(z*v)) by FUNCT_2:15
      .=g.(z*f.v) by CLOPBAN1:def 3
      .=z*g.(f.v) by CLOPBAN1:def 3
      .=z*(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,CLOPBAN1:def 3,VECTSP_1:def 20;
end;
