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 Th28:
  for R1,R2 holds R1+R2 is RestFunc of S,T & R1-R2 is RestFunc of S,T
proof
  let R1,R2;
A1: R1 is total & R2 is total by Def5;
A2: now
    let h ;
    assume A3: h is non-zero;
A4: (||.h.||")(#)(R1/*h) is convergent & (||.h.||")(#)(R2/*h) is
    convergent by Def5,A3;
A5: (||.h.||")(#)((R1+R2)/*h) = (||.h.||")(#)((R1/*h)+(R2/*h)) by A1,Th25
      .= ((||.h.||")(#)(R1/*h))+((||.h.||")(#)(R2/*h)) by Th9;
    hence (||.h.||")(#)((R1+R2)/*h) is convergent by A4,NORMSP_1:19;
    lim ((||.h.||")(#)(R1/*h)) = 0.T & lim ((||.h.||")(#)(R2/*h)) = 0.T
    by A3,Def5;
    hence lim ((||.h.||")(#)((R1+R2)/*h)) = 0.T+0.T by A4,A5,NORMSP_1:25
      .= 0.T by RLVECT_1:4;
  end;
A6: now
    let h;
    assume A7: h is non-zero;
A8: (||.h.||")(#)(R1/*h) is convergent & (||.h.||")(#)(R2/*h) is
    convergent by Def5,A7;
A9: (||.h.||")(#)((R1-R2)/*h) = (||.h.||")(#)((R1/*h)-(R2/*h)) by A1,Th25
      .= ((||.h.||")(#)(R1/*h))-((||.h.||")(#)(R2/*h)) by Th12;
    hence (||.h.||")(#)((R1-R2)/*h) is convergent by A8,NORMSP_1:20;
    lim ((||.h.||")(#)(R1/*h)) = 0.T & lim ((||.h.||")(#)(R2/*h)) = 0.T
    by Def5,A7;
    hence lim ((||.h.||")(#)((R1-R2)/*h)) = 0.T-0.T by A8,A9,NORMSP_1:26
      .= 0.T by RLVECT_1:13;
  end;
  R1+R2 is total by A1,VFUNCT_1:32;
  hence R1+R2 is RestFunc of S,T by A2,Def5;
  R1-R2 is total by A1,VFUNCT_1:32;
  hence thesis by A6,Def5;
end;
