reserve X, Y for RealNormSpace;

theorem Th6:
  for T be LinearOperator of X,Y, B0 be Subset of X, x1 be Point of
  X holds T.:(x1+ B0) = T.x1 + T.:B0
proof
  let T be LinearOperator of X,Y, B0 be Subset of X, x1 be Point of X;
  thus T.:(x1+ B0) c= T.x1 + T.:B0
  proof
    let t be object;
    assume t in T.:(x1+ B0);
    then consider z1 be object such that
A1: z1 in the carrier of X and
A2: z1 in x1+ B0 and
A3: t= T.z1 by FUNCT_2:64;
    reconsider z1 as Point of X by A1;
    consider z0 be Element of X such that
A4: z1=x1+ z0 and
A5: z0 in B0 by A2;
A6: T.z0 in T.:B0 by A5,FUNCT_2:35;
    t=T.x1 + T.z0 by A3,A4,VECTSP_1:def 20;
    hence thesis by A6;
  end;
  let t be object;
  assume t in T.x1 + T.:B0;
  then consider tz0 be Point of Y such that
A7: t=T.x1+tz0 and
A8: tz0 in T.:B0;
  consider z0 be Element of X such that
A9: z0 in B0 and
A10: tz0=T.z0 by A8,FUNCT_2:65;
A11: x1+z0 in x1+B0 by A9;
  t=T.(x1+z0) by A7,A10,VECTSP_1:def 20;
  hence thesis by A11,FUNCT_2:35;
end;
