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 Th5:
  r(#)R is RestFunc
proof
A1: R is total by Def2;
  now
    let h;
A2: (h")(#)((r(#)R)/*h) = (h")(#)(r(#)(R/*h)) by A1,RFUNCT_2:14
      .= r(#)((h")(#)(R/*h)) by SEQ_1:19;
A3: (h")(#)(R/*h) is convergent by Def2;
    hence (h")(#)((r(#)R)/*h) is convergent by A2;
    lim ((h")(#)(R/*h)) = 0 by Def2;
    hence lim ((h")(#)((r(#)R)/*h)) = r*0 by A3,A2,SEQ_2:8
      .= 0;
  end;
  hence thesis by A1,Def2;
end;
