reserve y for object, X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1 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 f,f1,f2 for PartFunc of REAL,REAL;
reserve h for non-zero 0-convergent Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc;
reserve L,L1,L2 for LinearFunc;

theorem Th7:
  R(#)L is RestFunc & L(#)R is RestFunc
proof
A1: L is total by Def3;
  consider x1 such that
A2: for p holds L.p=x1*p by Def3;
A3: R is total by Def2;
A4: now
    let h;
A5: (h")(#)(R/*h) is convergent by Def2;
    now
      let n;
      thus (L/*h).n=L.(h.n) by A1,FUNCT_2:115
        .=x1*(h.n) by A2
        .=(x1(#)h).n by SEQ_1:9;
    end;
    then
A6: (L/*h)=x1(#)h by FUNCT_2:63;
A7: L/*h is convergent by A6;
    lim h=0;
    then
A8: lim (L/*h)=x1*0 by A6,SEQ_2:8
      .=0;
A9: (h")(#)((R(#)L)/*h)=(h")(#)((R/*h)(#)(L/*h)) by A3,A1,RFUNCT_2:13
      .=((h")(#)(R/*h))(#)(L/*h) by SEQ_1:14;
    hence (h")(#)((R(#)L)/*h) is convergent by A7,A5;
    lim ((h")(#)(R/*h))=0 by Def2;
    hence lim ((h")(#)((R(#)L)/*h))=0*0 by A9,A7,A8,A5,SEQ_2:15
      .=0;
  end;
  hence R(#)L is RestFunc by A3,A1,Def2;
  thus thesis by A3,A1,A4,Def2;
end;
