
theorem Th9:
  for X,Y be RealNormSpace, T be LinearOperator of X,Y holds
  graphNSP(T) is Subspace of [:X,Y:]
proof
  let X,Y be RealNormSpace,
      T be LinearOperator of X,Y;
  set l = graphNSP(T);
  reconsider W = the RLSStruct of l as Subspace of [:X,Y:] by RSSPACE:11;
A1: 0.l = 0.W
       .= 0.([:X,Y:]) by RLSUB_1:def 2;
A2: the addF of l =(the addF of ([:X,Y:]) ) || (the carrier of W)
                 by RLSUB_1:def 2
              .= (the addF of ([:X,Y:])) || (the carrier of l);
  the Mult of l =(the Mult of ([:X,Y:])) | [:REAL, the carrier of W:]
              by RLSUB_1:def 2
                 .=(the Mult of ([:X,Y:]) ) | [:REAL, the carrier of l:];
  hence l is Subspace of [:X,Y:] by A1,A2,RLSUB_1:def 2;
end;
