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 Th6:
  L1(#)L2 is RestFunc-like
proof
  consider x1 such that
A1: for p holds L1.p=x1*p by Def3;
A2: L1 is total & L2 is total by Def3;
  hence L1(#)L2 is total;
  consider x2 such that
A3: for p holds L2.p=x2*p by Def3;
  now
    let h;
    now
      let n;
A4:   h.n<>0 by SEQ_1:5;
      thus ((h")(#)((L1(#)L2)/*h)).n=(h").n *((L1(#)L2)/*h).n by SEQ_1:8
        .=(h").n*(L1(#)L2).(h.n) by A2,FUNCT_2:115
        .=(h").n*(L1.(h.n)*L2.(h.n)) by A2,RFUNCT_1:56
        .=(h").n*L1.(h.n)*L2.(h.n)
        .=((h.n)")*L1.(h.n)*L2.(h.n) by VALUED_1:10
        .=((h.n)")*((h.n)*x1)*L2.(h.n) by A1
        .=((h.n)")*(h.n)*x1*L2.(h.n)
        .=1*x1*L2.(h.n) by A4,XCMPLX_0:def 7
        .=x1*(x2*(h.n)) by A3
        .=x1*x2*(h.n)
        .=((x1*x2)(#)h).n by SEQ_1:9;
    end;
    then
A5: (h")(#)((L1(#)L2)/*h)=(x1*x2)(#)h by FUNCT_2:63;
    thus (h")(#)((L1(#)L2)/*h) is convergent by A5;
    lim h=0;
    hence lim ((h")(#)((L1(#)L2)/*h)) = (x1*x2)*0 by A5,SEQ_2:8
      .=0;
  end;
  hence thesis;
end;
