reserve X, Y for RealNormSpace;

theorem Th5:
  for T be LinearOperator of X,Y, B0 be Subset of X, a be Real
  holds T.:(a*B0) = a*T.:B0
proof
  let T be LinearOperator of X,Y, B0 be Subset of X, a be Real;
  thus T.:(a*B0) c= a*T.:B0
  proof
    let t be object;
    assume t in T.:(a*B0);
    then consider z1 be object such that
    z1 in the carrier of X and
A1: z1 in a*B0 and
A2: t= T.z1 by FUNCT_2:64;
    consider x1 be Element of X such that
A3: z1=a*x1 and
A4: x1 in B0 by A1;
A5: T.x1 in T.:B0 by A4,FUNCT_2:35;
    t=a* T.x1 by A2,A3,LOPBAN_1:def 5;
    hence thesis by A5;
  end;
  let t be object;
  assume t in a*T.:B0;
  then consider tz0 be Point of Y such that
A6: t=a*tz0 and
A7: tz0 in T.:B0;
  consider z0 be Element of X such that
A8: z0 in B0 and
A9: tz0=T.z0 by A7,FUNCT_2:65;
  reconsider z0 as Point of X;
  set z1=a*z0;
A10: z1 in a*B0 by A8;
  t=T. z1 by A6,A9,LOPBAN_1:def 5;
  hence thesis by A10,FUNCT_2:35;
end;
