reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1 for sequence of S;
reserve x0 for Point of S;
reserve h for (0.S)-convergent sequence of S;
reserve c for constant sequence of S;
reserve R,R1,R2 for RestFunc of S,T;
reserve L,L1,L2 for Point of R_NormSpace_of_BoundedLinearOperators(S,T);

theorem Th29:
  for r,R holds r(#)R is RestFunc of S,T
proof
  let r,R;
A1: R is total by Def5;
A2: now
    let h;
    assume A3: h is non-zero;
A4: (||.h.||")(#)(R/*h) is convergent by A3,Def5;
A5: (||.h.||")(#)((r(#)R)/*h) = (||.h.||")(#)(r*(R/*h)) by A1,Th26
      .= r*((||.h.||")(#)(R/*h)) by Th10;
    hence (||.h.||")(#)((r(#)R)/*h) is convergent by A4,NORMSP_1:22;
    lim ((||.h.||")(#)(R/*h)) = 0.T by A3,Def5;
    hence lim ((||.h.||")(#)((r(#)R)/*h)) = r*0.T by A4,A5,NORMSP_1:28
      .= 0.T by RLVECT_1:10;
  end;
  r(#)R is total by A1,VFUNCT_1:34;
  hence thesis by A2,Def5;
end;
