reserve X, Y for RealNormSpace;

theorem
  for T be LinearOperator of X,Y, B0,B1 be Subset of X holds T.:(B0+B1)
  = T.:B0+T.:B1
proof
  let T be LinearOperator of X,Y, B0,B1 be Subset of X;
  thus T.:(B0+B1) c= T.:B0+T.:B1
  proof
    let t be object;
    assume t in T.:(B0+B1);
    then consider z1 be object such that
    z1 in the carrier of X and
A1: z1 in B0+B1 and
A2: t= T.z1 by FUNCT_2:64;
    consider x1,x2 be Element of X such that
A3: z1=x1+x2 and
A4: x1 in B0 & x2 in B1 by A1;
A5: T.x1 in T.:B0 & T.x2 in T.:B1 by A4,FUNCT_2:35;
    t=T.x1+T.x2 by A2,A3,VECTSP_1:def 20;
    hence thesis by A5;
  end;
  let t be object;
  assume t in T.:B0 + T.:B1;
  then consider tz0,tz1 be Point of Y such that
A6: t=tz0+tz1 and
A7: tz0 in T.:B0 and
A8: tz1 in T.:B1;
  consider z1 be Element of X such that
A9: z1 in B1 and
A10: tz1=T.z1 by A8,FUNCT_2:65;
  consider z0 be Element of X such that
A11: z0 in B0 and
A12: tz0=T.z0 by A7,FUNCT_2:65;
  reconsider z1 as Point of X;
  reconsider z0 as Point of X;
A13: z0+z1 in B0+B1 by A11,A9;
  t=T.(z0+z1) by A6,A12,A10,VECTSP_1:def 20;
  hence thesis by A13,FUNCT_2:35;
end;
