reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem Th7:
  R1+R2 is RestFunc of F & R1-R2 is RestFunc of F
  proof
    A1: R1 is total & R2 is total by Def1;
    A2: now
    let h;
    A3: (h")(#)((R1+R2)/*h) = (h")(#)((R1/*h)+(R2/*h)) by A1,Th6
    .= ((h")(#)(R1/*h))+((h")(#)(R2/*h)) by NDIFF_1:9;
    A4: (h")(#)(R1/*h) is convergent & (h")(#)(R2/*h) is convergent by Def1;
    hence (h")(#)((R1+R2)/*h) is convergent by A3,NORMSP_1:19;
    lim ((h")(#)(R1/*h)) = 0.F & lim ((h")(#)(R2/*h)) = 0.F by Def1;
    hence lim ((h")(#)((R1+R2)/*h)) = 0.F+0.F by A4,A3,NORMSP_1:25
    .= 0.F by RLVECT_1:def 4;
  end;
  R1+R2 is total by A1,VFUNCT_1:32;
  hence R1+R2 is RestFunc of F by A2,Def1;
  A5: now let h;
  A6: (h")(#)((R1-R2)/*h) = (h")(#)((R1/*h)-(R2/*h)) by A1,Th6
  .= ((h")(#)(R1/*h))-((h")(#)(R2/*h)) by NDIFF_1:12;
  A7: (h")(#)(R1/*h) is convergent & (h")(#)(R2/*h) is convergent by Def1;
  hence (h")(#)((R1-R2)/*h) is convergent by A6,NORMSP_1:20;
  lim ((h")(#)(R1/*h)) = 0.F & lim ((h")(#)(R2/*h)) = 0.F by Def1;
  hence lim ((h")(#)((R1-R2)/*h)) = 0.F-0.F by A7,A6,NORMSP_1:26
  .= 0.F by RLVECT_1:13;
end;
R1-R2 is total by A1,VFUNCT_1:32;
hence R1-R2 is RestFunc of F by A5,Def1;
end;
